Assertion (A) : In an experiment of throwing an unbiased die, the probability of getting a prime number given that number appearing on the die being odd is $\\frac{2}{3}$.
Reason (R) : For any two events A and B, $P(A|B) = \\frac{P(A \cup B)}{P(B)}$
CBSE CLASS 12 • MATHEMATICS • CHAPTER 13
Probability
212 distinct questions across 307 board paper set appearances (2022–2026).
(a) Out of two bags, bag I contains $3$ red and $4$ white balls and bag II contains $8$ red and $6$ white balls. A die is thrown. If it shows a number less than $3$ then a ball is drawn at random from bag I, otherwise a ball is drawn at random from bag II. Find the probability that the ball drawn fr...
In an online jackpot, there is one first prize of ₹ $3,00,000$, two second prizes of ₹ $2,00,000$ each and three third prizes of ₹ $50,000$ each.
A total of $1,00,000$ jackpot tickets each costing ₹ $100$ were sold there by raising a fund of ₹ $1,00,00,000$.
Rohan bought one ticket.
Based on give...
(a) The probability of hitting the target by a trained sniper is three times the probability of not hitting the target on a stormy day due to high wind speed.\nThe sniper fired two shots on the target on a stormy day when wind speed was very high. Find the probability that\n(i) target is hit\n(ii) a...
In a school, the probability of holding a debate competition is $\frac{1}{3}$ and that of a quiz competition is $\frac{2}{3}$. In the two participating teams, A has 4 girls and 6 boys and B has 7 girls and 3 boys. If a debate competition is held, the students are selected from team A and for the qui...
An NGO organises a charity event in which they decide to distribute woollen caps to protect children from winter. The caps to be distributed are in three separate boxes, Box I has 30 red caps, Box II has 20 red and 10 green caps, and Box III has 30 green caps. The probability that a Box i is selecte...
There are three types of vaccines A$_1$, A$_2$, A$_3$, available in the market to protect the population of the country from spread of certain infection. According to a survey conducted, it was found that 25% of the population was given Vaccine A$_1$, 35% of the population was given Vaccine A$_2$ an...
(a) A box contains cards numbered $1$ to $22$. Two cards are drawn simultaneously without replacement and their numbers are noted. What is the probability that both numbers are even ?
OR
(b) If $\text{P}(A) = \frac{3}{8}$, $\text{P}(B) = \frac{1}{2}$ and $\text{P}(A \cap B) = \frac{1}{4}$, find $\te...
A coin is tossed three times. The probability of getting at least two heads is :
(A) $\frac{1}{2}$
(B) $\frac{3}{8}$
(C) $\frac{1}{8}$
(D) $\frac{1}{4}$
(a) Four students of class XII are given a problem to solve independently. Their chances of solving the problem respectively are $\frac{1}{2}, \frac{1}{3}, \frac{2}{3}$ and $\frac{1}{5}$. Find the probability that at most one of them will solve the problem.\nOR\n(b) The probability distribution of a...
Case Study - 3
Two persons are competing for a position on the Managing Committee of an organisation. The probabilities that the first and the second person will be appointed are $0\cdot5$ and $0\cdot6$ respectively. Also, if the first person gets appointed, then the probability of introducing wast...
(a) The probability that a student buys a colouring book is $0\cdot7$ and that she buys a box of colours is $0\cdot2$. The probability that she buys a colouring book, given that she buys a box of colours, is $0\cdot3$. Find the probability that the student :
(i) Buys both the colouring book and the...
A coin is tossed and a card is selected at random from a well shuffled pack of 52 playing cards. The probability of getting head on the coin and a face card from the pack is :
(A) $\frac{2}{13}$
(B) $\frac{3}{26}$
(C) $\frac{19}{26}$
(D) $\frac{3}{13}$
(a) 10 identical blocks are marked with '0' on two of them, '1' on three of them, '2' on four of them and '3' on one of them and put in a box. If $X$ denotes the number written on the block, then write the probability distribution of $X$ and calculate its mean.
OR
(b) In a village of 8000 people, ...
A pair of dice is thrown simultaneously. If $X$ denotes the absolute difference of the numbers appearing on top of the dice, then find the probability distribution of $X$.
The chances of P, Q and R getting selected as CEO of a company are in the ratio $4 : 1 : 2$ respectively. The probabilities for the company to increase its profits from the previous year under the new CEO, P, Q or R are $0\cdot3$, $0\cdot8$ and $0\cdot5$ respectively. If the company increased the pr...
(a) A card from a well shuffled deck of 52 playing cards is lost. From the remaining cards of the pack, a card is drawn at random and is found to be a King. Find the probability of the lost card being a King. OR (b) A biased die is twice as likely to show an even number as an odd number. If such a d...
A departmental store sends bills to charge its customers once a month. Past experience shows that 70% of its customers pay their first month bill in time. The store also found that the customer who pays the bill in time has the probability of 0.8 of paying in time next month and the customer who doe...
(a) Out of two bags, bag A contains 2 white and 3 red balls and bag B contains 4 white and 5 red balls. One ball is drawn at random from one of the bags and is found to be red. Find the probability that it was drawn from bag B. OR (b) Out of a group of 50 people, 20 always speak the truth. Two perso...
In a group activity class, there are 10 students whose ages are 16, 17, 15, 14, 19, 17, 16, 19, 16 and 15 years. One student is selected at random such that each has equal chance of being chosen and age of the student is recorded. On the basis of the above information, answer the following questions...
Two balls are drawn at random from a bag containing 2 red balls and 3 blue balls, without replacement. Let the variable X denotes the number of red balls. Find the probability distribution of X.
A pair of dice is thrown. It is given that the sum of numbers appearing on both dice is an even number. Find the probability that the number appearing on at least one die is 3.
Probabilities of A and B solving a specific problem are $\frac{2}{3}$ and $\frac{3}{5}$, respectively. If both of them try independently to solve the problem, then find the probability that the problem is solved.
Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of spade cards.
A pair of dice is thrown and the sum of the numbers appearing on the dice is observed to be 7. Find the probability that the number 5 has appeared on atleast one die.
In a game of Archery, each ring of the Archery target is valued. The centremost ring is worth 10 points and rest of the rings are allotted points 9 to 1 in sequential order moving outwards. Archer A is likely to earn 10 points with a probability of 0.8 and Archer B is likely the earn 10 points with ...
A bag contains cards numbered 1 to 25. Two cards are drawn at random, one after the other, without replacement. Find the probability that the number on each card is a multiple of 7.
One bag contains 4 white and 5 black balls. Another bag contains 6 white and 7 black balls. A ball, drawn at random, is transferred from the first bag to the second bag and then a ball is drawn at random from the second bag. Find the probability that the ball drawn is white.
A biased die is tossed and respective probabilities for various faces to turn up are the following :
| Face | 1 | 2 | 3 | 4 | 5 | 6 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| Probability | 0.1 | 0.24 | 0.19 | 0.18 | 0.15 | K |
Based on the above information, answer the following quest...
Smoking increases the risk of lung problems.
A study revealed that $170$ in $1000$ males who smoke develop lung complications, while $120$ out of $1000$ females who smoke develop lung related problems. In a colony, $50$ people were found to be smokers of which $30$ are males.
A person is selected at...
A survey was conducted on the patients who have undergone knee replacement surgeries.
It was found that, Robotic Knee replacement surgeries have $90%$ success rate.
On a particular day, robotic surgery was performed on three patients, A, B and C, one after the other. Assuming that the success and f...
If $3P(A) = P(B) = \frac{3}{5}$ and $P(A|B) = \frac{2}{3}$, then $P(A \cup B)$ is :
(A) $\frac{3}{5}$
(B) $\frac{1}{5}$
(C) $\frac{2}{15}$
(D) $\frac{2}{5}$
An organisation needs to select a building construction company to make houses in cities. Three companies X, Y and Z are short-listed with the probability of their selection being $\frac{1}{3}$, $\frac{2}{9}$ and $\frac{4}{9}$ respectively. It was found that there were $20%$, $40%$ and $30%$ chances...
A bank offers loan to its customers on different types of interest namely, fixed rate, floating rate and variable rate. From the past data with the bank, it is known that a customer avails loan on fixed rate, floating rate or variable rate with probabilities $10%$, $20%$ and $70%$ respectively. A cu...
(a) Find the probability distribution of the number of boys in families having three children, assuming equal probability for a boy and a girl. OR (b) A coin is tossed twice. Let X be a random variable defined as number of heads minus number of tails. Obtain the probability distribution of X and als...
What is the probability of a student not having misconception but still answers Bijoy in the test?
What is the probability that a randomly selected student answers Bijoy as his answer in the test?
What is the probability that a student who answered as Bijoy is having misconception?
What is the probability that a student who answered as Bijoy is amongst students who do not have the misconception?
Assertion (A) : If A and B are two events such that $P(A \cap B) = 0$, then A and B are independent events. Reason (R) : Two events are independent if the occurrence of one does not effect the occurrence of the other.
A gardener wanted to plant vegetables in his garden. Hence he bought 10 seeds of brinjal plant, 12 seeds of cabbage plant and 8 seeds of radish plant. The shopkeeper assured him of germination probabilities of brinjal, cabbage and radish to be 25%, 35% and 40% respectively. But before he could plant...
A shop selling electronic items sells smartphones of only three reputed companies A, B and C because chances of their manufacturing a defective smartphone are only $5%$, $4%$ and $2%$ respectively. In his inventory he has $25%$ smartphones from company A, $35%$ smartphones from company B and $40%$ s...
Based upon the results of regular medical check-ups in a hospital, it was found that out of 1000 people, 700 were very healthy, 200 maintained average health and 100 had a poor health record. Let $A_1$ : People with good health, $A_2$ : People with average health, and $A_3$ : People with poor health...
Self-study helps students to build confidence in learning. It boosts the self-esteem of the learners. Recent surveys suggested that close to 50% learners were self-taught using internet resources and upskilled themselves. A student may spend 1 hour to 6 hours in a day in upskilling self. The probabi...
(i) Express the probability distribution given above in the form of a probability distribution table.
The probability distribution of a random variable X is :
| X | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| P(X) | 0.1 | k | 2k | k | 0.1 |
where k is some unknown constant.
The probability that the random variable X takes the value 2 is :
(A) $\frac{1}{5}$
(B) $\frac{2}{5}$
(C) $\frac{4}{5}$
(...
(i) What is the probability that at least one of them is selected ?
(iii) Find the probability that exactly one of them is selected.
The random variable X has the following probability distribution where a and b are some constants :
| X | 1 | 2 | 3 | 4 | 5 |
| :--- | :--- | :--- | :--- | :--- | :--- |
| P(X) | 0.2 | a | a | 0.2 | b |
If the mean E(X) = 3, then find values of a and b and hence determine P(X $\geq$ 3).
A departmental store sends bills to charge its customers once a month. Past experience shows that $70%$ of its customers pay their first month bill in time. The store also found that the customer who pays the bill in time has the probability of $0.8$ of paying in time next month and the customer who...
Bag I contains 3 red and 4 black balls, Bag II contains 5 red and 2 black balls. Two balls are transferred at random from Bag I to Bag II and then a ball is drawn at random from Bag II. Find the probability that the drawn ball is red in colour.
An urn contains 3 red and 2 white marbles. Two marbles are drawn one by one with replacement from the urn. Find the probability distribution of the number of white balls. Also, find the mean of the number of white balls drawn.
It is known that 20% of the students in a school have above 90% attendance and 80% of the students are irregular. Past year results show that 80% of students who have above 90% attendance and 20% of irregular students get ‘A’ grade in their annual examination. At the end of a year, a student is chos...
(iii) (b) Find the probability that a randomly selected person of the society does Yoga of type A or B but not C.
A and B are independent events such that $P(A \cap \bar{B}) = \frac{1}{4}$ and $P(\bar{A} \cap B) = \frac{1}{6}$. Find $P(A)$ and $P(B)$.
(a) Two balls are drawn at random one by one with replacement from an urn containing equal number of red balls and green balls. Find the probability distribution of number of red balls. Also, find the mean of the random variable.
OR
(b) A and B throw a die alternately till one of them gets a '6' and...
(a) A pair of dice is thrown simultaneously. If X denotes the absolute difference of numbers obtained on the pair of dice, then find the probability distribution of X.
OR
(b) There are two coins. One of them is a biased coin such that P (head) : P (tail) is 1 : 3 and the other coin is a fair coin....
An octagonal prism is a three-dimensional polyhedron bounded by two octagonal bases and eight rectangular side faces. It has 24 edges and 16 vertices. The prism is rolled along the rectangular faces and number on the bottom face (touching the ground) is noted. Let X denote the number obtained on the...
From a lot of 30 bulbs which include 6 defective bulbs, a sample of 2 bulbs is drawn at random one by one with replacement. Find the probability distribution of the number of defective bulbs and hence find the mean number of defective bulbs.
A building contractor undertakes a job to construct 4 flats on a plot along with parking area. Due to strike the probability of many construction workers not being present for the job is 0.65. The probability that many are not present and still the work gets completed on time is 0.35. The probabilit...
(a) In answering a question on a multiple choice test, a student either knows the answer or guesses. Let $\frac{3}{5}$ be the probability that he knows the answer and $\frac{2}{5}$ be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability ...
A family has 2 children and the elder child is a girl. The probability that both children are girls is :
(a) $\frac{1}{4}$
(b) $\frac{1}{8}$
(c) $\frac{1}{2}$
(d) $\frac{3}{4}$
A card from a pack of $52$ playing cards is lost. From the remaining cards, $2$ cards are drawn at random without replacement, and are found to be both aces. Find the probability that lost card being an ace.
OR
The probability that A hits the target is $\frac{1}{3}$ and the probability that B hits it, is $\frac{2}{5}$. If both try to hit the target independently, find the probability that the target is hit.
Case Study Problem : A shopkeeper sells three types of flower seeds A1, A2, A3. They are sold in the form of a mixture, where the proportions of these seeds are $4 : 4 : 2$, respectively. The germination rates of the three types of seeds are $45%$, $60%$ and $35%$ respectively. Based on the above in...
A coin is tossed twice. The following table shows the probability distribution of number of tails :
| X | 0 | 1 | 2 |
|---|---|---|---|
| P(X) | K | 6K | 9K |
(a) Find the value of $K$.
(b) Is the coin tossed biased or unbiased ? Justify your answer.
What is the probability that the person does not win any amount ?
In another jackpot, Rohan also bought a ticket having a prize money of ₹ 5,00,000. The chances of winning the jackpot are $1$ in $1,00,000$. Find the probability that on exactly one of tickets he wins the jackpot.
A study revealed that 170 in 1000 males who smoke develop lung complications, while 120 out of 1000 females who smoke develop lung related problems. In a colony, 50 people were found to be smokers of which 30 are males.\nA person is selected at random from these 50 people and tested for lung related...
If events $A$ and $B$ are such that $P(A|B) = P(B|A)$, then :
(A) $A \subset B$ but $A \neq B$
(B) $A = B$
(C) $A \cap B = \phi$
(D) $P(A) = P(B)$
In a family with three children, the probability that the eldest child is a boy, given that the family has at least one boy, is :
(A) $\frac{1}{2}$
(B) $\frac{1}{3}$
(C) $\frac{2}{3}$
(D) $\frac{4}{7}$
Assertion (A) : For two independent events A and B, P(A \mid B) = P(A). \n\n Reason (R) : A and B are independent events, then P(A \cap B) = P(A) \cdot P(B).
Two bags contain balls such that Bag I contains $5$ red and $6$ black balls, while Bag II contains $3$ red and $4$ black balls.
Based on the above information, answer the following questions :
(i) One ball is drawn at random from each of the two bags. Find the probability of getting both red balls...
If E and F are two independent events such that $\text{P}(E) = \frac{3}{10}$, $\text{P}(E \cup F) = \frac{1}{2}$, then $\text{P}(E|F) - \text{P}(F|E)$ is equal to :
A survey was conducted on the patients who have undergone knee replacement surgeries. It was found that robotic knee replacement surgery is $90%$ successful.
On a particular day, three robotic surgery patients $A$, $B$ and $C$ underwent the surgery one after the other. Assuming that the success and...
It was found that, Robotic Knee replacement surgeries have $90%$ success rate.
On a particular day, robotic surgery was performed on three patients, A, B and C, one after the other. Assuming that the success and failure of each surgery is independent of each other, find the probability that :
(i) ex...
A survey was conducted to find out the success rate of students who qualified the entrance examination by dropping a year after class XII.
As per the data collected, $40%$ students appearing in the examination were dropouts and the remaining students were regular students of class XII.
Of the drop...
If $E$ and $F$ are two independent events such that $P(E) = \frac{3}{10}$, $P(E \cup F) = \frac{1}{2}$, then $P(E|F) - P(F|E)$ is equal to :
(A) $\frac{2}{7}$
(B) $\frac{3}{35}$
(C) $\frac{1}{70}$
(D) $\frac{1}{7}$
As per the data collected, $40%$ students appearing in the examination were dropouts and the remaining students were regular students of class XII.
Of the dropouts, $5%$ qualify the examination while $10%$ of the regular students qualify the examination.
Based on the given information, answer the ...
Based on the above information, answer the following questions:
(i) Find the probability that a student selected at random is a regular student.
(ii) A student is selected at random from a group of dropout students. What is the probability that the student will not qualify the examination ?
(iii) (a...
If $E$ and $F$ are two independent events such that $P(E) = \frac{3}{10}$, $P(E \cup F) = \frac{1}{2}$, then $P(E \mid F) - P(F \mid E)$ is equal to :
(A) $\frac{2}{7}$
(B) $\frac{3}{35}$
(C) $\frac{1}{70}$
(D) $\frac{1}{7}$
A survey was conducted on patients who underwent knee replacement surgery. It was found that robotic knee replacement surgery is $90%$ successful. On a particular day, three robotic surgery patients A, B and C were operated one after the other. Assuming that the success and failure of each surgery a...
Case Study - 3
A survey was conducted to find out the success rate of students who qualified the entrance examination by dropping a year after class XII.
As per the data collected, 40% students appearing in the examination were dropouts and the remaining students were regular students of class XII.
...
A box contains 4 red, 5 blue and 1 green marble. A child randomly takes out a marble from the box, notes down the colour and puts it back in the box. If the activity is repeated 3 times, what is the probability that at least one marble is red ?
(A) $\frac{27}{125}$
(B) $\frac{8}{125}$
(C) $\frac{2}{...
The probability that it will rain tomorrow in cities A, B and C is $60%, 70%$ and $80%$ respectively. The probability that it will rain tomorrow in at least one of the cities is :
(A) $\frac{3}{250}$
(B) $\frac{244}{250}$
(C) $1$
(D) $\frac{9}{10}$
The probability that a particular item is available in three shops A, B and C is $\frac{4}{5}$, $\frac{3}{4}$ and $\frac{2}{3}$ respectively. If a person visits all the three shops to buy the item, then what is the probability that it will be available in at least one of the shops ?
(A) $\frac{59}{6...
(b) A box contains 6 cards numbered 1 to 6. A student is asked to pick up two cards, one by one after replacement and note down the numbers on the cards. Let A be the event of getting sum of the numbers on two cards as 10, and B, the event of a number other than 4 on the first card selected. Find P(...
If $3\text{P(A)} = \text{P(B)} = \frac{3}{5}$ and $\text{P(A|B)} = \frac{2}{3}$, then $\text{P(A} \cup \text{B)}$ is :
(A) $\frac{3}{5}$
(B) $\frac{1}{5}$
(C) $\frac{2}{15}$
(D) $\frac{2}{5}$
If the probability distribution of a discrete random variable X is given as :
\begin{tabular}{|l|c|c|c|c|}
\hline
X & 1 & 2 & 3 & 4 \\
\hline
P(X) & $\frac{1}{10}$ & $\frac{1}{5}$ & $\frac{3}{10}$ & $\frac{2}{5}$ \\
\hline
\end{tabular}
then, $E(X^2)$ is equal to :
(A) 3
(B) 5
(C) 7
(D) 10
If the variance of a Poisson distribution is 2, then $P(X = 2)$ is :
(A) $\frac{2}{e^2}$
(B) $2e^2$
(C) $\frac{4}{e^2}$
(D) $4e^2$
In a binomial distribution, the probability of getting a success is $\frac{1}{4}$ and the standard deviation is $3$. Then, its mean is :
(A) $6$
(B) $8$
(C) $12$
(D) $48$
Assertion ($A$) : In a binomial distribution, if mean $= 4$ and variance $= 3$, then the number of trials is $16$.
Reason ($R$) : In a binomial distribution with parameters $n$ and $p$, mean $= np$ and variance $= \sqrt{np}$ .
According to an educational board survey, it was observed that class XII students apply at least one to four weeks before the colleges' application deadlines and the probability of a student to get admission in the college $ ext{P}( ext{X} = ext{x})$ is given as follows :
$$
\text{P}(\text{X} = \te...
A coin is tossed $3$ times. The probability of getting at least one head is :
(A) $\frac{1}{8}$
(B) $\frac{3}{8}$
(C) $\frac{7}{8}$
(D) $\frac{1}{2}$
A die is thrown once and a card is selected at random from a pack of $52$ playing cards. The probability of getting a number greater than $4$ and a card of Hearts is :
(A) $\frac{1}{3}$
(B) $\frac{1}{4}$
(C) $\frac{7}{12}$
(D) $\frac{1}{12}$
A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn randomly and are found to be both Spades. Find the probability of the lost card being a Spade.
If $A$ and $B$ are independent events such that $P(A) = 0.45$ and $P(A \cup B) = 0.5$, then $P(B)$ is equal to :
(A) $\frac{1}{29}$
(B) $\frac{10}{11}$
(C) $\frac{10}{29}$
(D) $\frac{1}{11}$
Case Study - 3
An organisation needs to select a building construction company to make houses in cities. Three companies X, Y and Z are short-listed with the probability of their selection being $\frac{1}{3}$, $\frac{2}{9}$ and $\frac{4}{9}$ respectively. It was found that there were $20%$, $40%$ a...
A person observed the first 4 digits of your 6-digit PIN. What is the probability that the person can guess your PIN?
(A) $\frac{1}{81}$
(B) $\frac{1}{100}$
(C) $\frac{1}{90}$
(D) $1$
Two students Mehul and Rashi are seeking admission in a college. The probability that Mehul is selected is $0.4$ and the probability of selection of exactly one of the them is $0.5$. Chances of selection of them is independent of each other. Find the chances of selection of Rashi. Also find the prob...
Case Study -3
Excessive use of screens can result in vision problems, obesity, sleep disorders, anxiety, low retention problems and can impede social and emotional comprehension and expression. It is essential to be mindful of the amount of time we spend on screens and to reduce our screen-time by ...
For a random variable $X$, $E(X) = 3$ and $E(X^2) = 11$. The variance of $X$ is :
(A) $8$
(B) $5$
(C) $2$
(D) $1$
If the mean and standard deviation of a binomial distribution are $12$ and $2$ respectively, then the value of the parameter $p$ is :
(A) $\frac{5}{6}$
(B) $\frac{1}{6}$
(C) $\frac{1}{3}$
(D) $\frac{2}{3}$
If the variance of a Poisson distribution is $2$, then $P(X = 2)$ is :
(A) $4\text{e}^2$
(B) $2\text{e}^2$
(C) $\frac{2}{\text{e}^2}$
(D) $\frac{4}{\text{e}^2}$
Normal distribution is symmetric about :
(A) Variance
(B) Co-variance
(C) Mean
(D) Standard deviation
Find the probability distribution of a number of successes in two tosses of a die, where a success is defined as getting a number greater than $4$.
According to an educational board survey, it was observed that class XII students apply at least one to four weeks ahead of college application deadlines. Let X represent the week when an average student applies ahead of a college's application deadline and the probability of the student to get admi...
What is the probability that Mahesh will get admission in the college, given that he applied at least 3 weeks ahead of application deadline ?
Two balls are drawn at random without replacement from a box containing $3$ black and $7$ red balls. Find the probability that :
(i) both balls are red.
(ii) first ball is black and the second is red.
Two cards are drawn successively with replacement of a well-shuffled pack of $52$ playing cards. Find the probability distribution of the number of face cards.
Case Study - 3
A coach is training $3$ players. He observes that player A can hit the target $4$ times in $5$ shots, player B can hit the target $3$ times in $4$ shots and player C can hit the target $2$ times in $3$ shots.
Based on the above information, answer the following questions :
If they al...
E and F are two independent events such that $P(\overline{E}) = 0\cdot6$ and $P(E \cup F) = 0\cdot6$. Find $P(F)$ and $P(\overline{E} \cup \overline{F})$.
If E and F are two independent events such that $\text{P(E)} = \frac{2}{3}$, $\text{P(F)} = \frac{3}{7}$, then $\text{P(E/\bar{F})}$ is equal to :
(A) $\frac{1}{6}$
(B) $\frac{1}{2}$
(C) $\frac{2}{3}$
(D) $\frac{7}{9}$
If E and F are two independent events such that $P(E) = \frac{2}{3}$, $P(F) = \frac{3}{7}$, then $P(E / \overline{F})$ is equal to :
(A) $\frac{1}{6}$
(B) $\frac{1}{2}$
(C) $\frac{2}{3}$
(D) $\frac{7}{9}$
(a) The probability distribution for the number of students being absent in a class on a Saturday is as follows :
| X | 0 | 2 | 4 | 5 |
| P(X) | p | 2p | 3p | p |
Where X is the number of students absent.
(i) Calculate p.
(ii) Calculate the mean of the number of absent students on Saturday.
OR
(b) F...
(i) What is the probability that a customer after availing the loan will default on the loan repayment?
(ii) A customer after availing the loan, defaults on loan repayment. What is the probability that he availed the loan at a variable rate of interest?
(a) The probability distribution for the number of students being absent in a class on a Saturday is as follows :
| $\mathbf{X}$ | $0$ | $2$ | $4$ | $5$ |
| $\mathbf{P(X)}$ | $\text{p}$ | $2\text{p}$ | $3\text{p}$ | $\text{p}$ |
Where X is the number of students absent.
(i) Calculate $\text{p}$.
(ii...
(a) A die with number 1 to 6 is biased such that $P(2) = \frac{3}{10}$ and probability of other numbers is equal. Find the mean of the number of times number 2 appears on the dice, if the dice is thrown twice.
OR
(b) Two dice are thrown. Defined are the following two events $A$ and $B$:
$A = \{(x, y...
In the following probability distribution, the value of p is :
| X | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| P(X) | p | p | 0.3 | 2p |
(A) $\frac{7}{40}$
(B) $\frac{1}{10}$
(C) $\frac{9}{35}$
(D) $\frac{1}{4}$
(a) A die with number $1$ to $6$ is biased such that $\text{P}(2) = \frac{3}{10}$ and probability of other numbers is equal. Find the mean of the number of times number $2$ appears on the dice, if the dice is thrown twice.
OR
(b) Two dice are thrown. Defined are the following two events $A$ and $B...
A and B appeared for an interview for two vacancies. The probability of A's selection is $\frac{1}{5}$ and that of B's selection is $\frac{1}{3}$. The probability that none of them is selected is :
(A) $\frac{11}{15}$
(B) $\frac{7}{15}$
(C) $\frac{8}{15}$
(D) $\frac{1}{5}$
(a) Find the probability distribution of the number of doublets in three throws of a pair of dice.
OR
(b) If $E$ and $F$ are two independent events with $P(E) = p$, $P(F) = 2p$ and $P(\text{exactly one of } E, F) = \frac{5}{9}$, then find the value of $p$.
Assertion (A) : If A and B are two events such that $\text{P(A} \cap \text{B)} = 0$, then A and B are independent events. Reason (R) : Two events are independent if the occurrence of one does not effect the occurrence of the other.
(a) Consider the experiment of tossing a coin. If the coin shows head, toss it again; but if it shows a tail, then throw a die. Find the conditional probability of the event A : ‘the die shows a number greater than 3’ given that B : ‘there is at least one tail’.
OR
(b) The probability distribution o...
Some students are having a misconception while comparing decimals. For example, a student may mention that $78.56 > 78.9$ as $7856 > 789$. In order to assess this concept, a decimal comparison test was administered to the students of class VI through the following question :
In the recently held Spo...
(a) The probability distribution of a random variable $X$ is given by :
| X | 0 | 1 | 2 | 3 |
| P(X) | p | $\frac{p}{3}$ | $\frac{p}{6}$ | $\frac{p}{12}$ |
(i) Determine the value of $p$.
(ii) Calculate $P(X \ge 1)$.
(iii) Calculate expectation of $X$, i.e. $E(X)$
OR
(b) In a city, a survey was cond...
Calculate the probability of a randomly chosen seed to germinate.
What is the probability that it is a cabbage seed, given that the chosen seed germinates ?
A box has 4 green, 8 blue and 3 red pens. A student picks up a pen at random, checks its colour and replaces it in the box. He repeats this process 3 times. The probability that at least one pen picked was red is :
(A) $\frac{124}{125}$
(B) $\frac{1}{125}$
(C) $\frac{61}{125}$
(D) $\frac{64}{125}$
A person is Head of two independent selection committees $\text{I}$ and $\text{II}$. If the probability of making a wrong selection in committee $\text{I}$ is $0\cdot03$ and that in committee $\text{II}$ is $0\cdot01$, then find the probability that the person makes the correct decision of selection...
Find the probability that it was defective.
What is the probability that this defective smartphone was manufactured by company B ?
Chances that three persons A, B, and C go to the market are $30%, 60%$ and $50%$ respectively. The probability that at least one will go to the market is :
(A) $\frac{14}{10}$
(B) $\frac{43}{50}$
(C) $\frac{9}{100}$
(D) $\frac{7}{50}$
A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed three times, find the probability distribution of number of tails. Hence, find the mean of the distribution.
A meeting will be held only if all three members A, B and C are present. The probability that member A does not turn up is $0\cdot10$, member B does not turn up is $0\cdot20$ and member C does not turn up is $0\cdot05$. The probability of the meeting being cancelled is :
(A) $0\cdot35$
(B) $0\cdot31...
Bag I contains $4$ white and $5$ black balls. Bag II contains $6$ white and $7$ black balls. A ball drawn randomly by from bag I is transferred to bag II and then a ball is drawn randomly from bag II. Find the probability that the ball drawn is white.
A person was tested randomly. What is the probability that he/she has contracted the disease?
Given that the person has not contracted the disease, what is the probability that the person is from category $A_2$?
E and F are two independent events such that $P(\bar{E}) = 0\cdot 6$ and $P(E \cup F) = 0\cdot 6$. Find $P(F)$ and $P(\bar{E} \cup \bar{F})$.
The probability that it rains today is $0.4$. If it rains today, the probability that it will rain tomorrow is $0.8$. If it does not rain today, the probability that it will rain tomorrow is $0.7$. If
$P_1$: denotes the probability that it does not rain today.
$P_2$: denotes the probability that it ...
A random variable $X$ can take all non-negative integral values and the probability that $X$ takes
When two birds flew from Cage-I to Cage-II and two birds flew back from Cage-II to Cage-I then find the probability that the owl is still in Cage-I.
When two birds flew from Cage-I to Cage-II and two birds flew back from Cage-II to Cage-I, the owl is still seen in Cage-I, what is the probability that one parrot and the owl flew from Cage-I to Cage-II?
The probabilities of A, B and C solving a problem are $\frac{1}{3}, \frac{1}{5}$ and $\frac{1}{6}$ respectively. The probability that the problem is solved, is :
(A) $\frac{4}{9}$
(B) $\frac{5}{9}$
(C) $\frac{1}{90}$
(D) $\frac{1}{3}$
(a) A card is randomly drawn from a well-shuffled pack of 52 playing cards. Events A and B are defined as under : A : Getting a card of diamond B : Getting a queen Determine whether the events A and B are independent or not. OR (b) Find the probability distribution of the number of doublets in three...
A coach is training 3 players. He observes that player A can hit a target 4 times in 5 shots, player B can hit 3 times in 4 shots and player C can hit 2 times in 3 shots. Based on the above, answer the following questions : (i) Find the probability that all three players miss the target.
(ii) Find the probability that all of them hit the target.
(iii) (a) Find the probability that only one of them hits the target.
(iii) (b) Find the probability that exactly two of them hit the target.
OR (iii) Find the probability that exactly two of them are selected.
The probability distribution of a random variable X is :
\begin{tabular}{|l|c|c|c|c|c|}
\hline
\textbf{X} & 0 & 1 & 2 & 3 & 4 \\ \hline
\textbf{P(X)} & 0.1 & k & 2k & k & 0.1 \\ \hline
\end{tabular}
where k is some unknown constant.
The probability that the random variable X takes the value 2 is :
(...
A biased die is twice as likely to show an even number as an odd number. If such a die is thrown twice, find the probability distribution of the number of sixes. Also, find the mean of the distribution.
Find the probability that exactly two of them are selected.
If A and B are events such that $P(A|B) = P(B|A) \neq 0$, then :
(A) $A \subset B$, but $A \neq B$
(B) $A = B$
(C) $A \cap B = \phi$
(D) $P(A) = P(B)$
A random variable $X$ takes the values $-1, 0, 1$. If its mean is $0.6$ and $P(X = 0) = 0.2$, then $P(X = 1)$ is : (A) 0.7 (B) 0.5 (C) 0.4 (D) 0.3
One hundred identical coins each with probability $p$ showing up heads are tossed once. If $0 < p < 1$ and the probability of heads on 50 coins is equal to that of heads showing on 51 coins, then the value of $p$ is : (A) $\frac{1}{2}$ (B) $\frac{49}{101}$ (C) $\frac{50}{101}$ (D) $\frac{51}{101}$
The probability that a bomb dropped from a plane strikes the target is $\frac{4}{5}$. What is the probability that out of 6 bombs dropped, exactly 2 bombs strike the target ? (A) $2(\frac{4}{5})^5$ (B) $1 - 2(\frac{4}{5})^5$ (C) $\frac{48}{5^5}$ (D) $\frac{64}{5^6}$
A specific characteristic of a sample is known as a : (A) population (B) parameter (C) statistic (D) variance
(a) An unbiased die is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the die.
(a) The probability distribution of a random variable X is given below :
X | 1 | 2 | 3
--- | --- | --- | ---
P(X) | k/2 | k/3 | k/6
(i) Find the value of k.
(ii) Find P(1 ≤ X < 3).
(iii) Find E(X), the mean of X.
OR
(b) A and B are independent events such that P(A ∩ Ī) = 1/4 and P(Ā ∩ B)...
The probability distribution of a random variable X is given below :
| X | 1 | 2 | 3 |
| P(X) | $\frac{k}{2}$ | $\frac{k}{3}$ | $\frac{k}{6}$ |
(i) Find the value of k.
(ii) Find $P(1 \le X < 3)$.
(iii) Find $E(X)$, the mean of X.
The Venn diagram below represents the probabilities of three different types of Yoga, A, B and C performed by the people of a society. Further, it is given that probability of a member performing type C Yoga is 0.44. On the basis of the above information, answer the following questions : (i) Find th...
Ashima can hit a target $2$ out of $3$ times. She tried to hit the target twice.
The probability that she missed the target exactly once is
(a) $\frac{2}{3}$
(b) $\frac{1}{3}$
(c) $\frac{4}{9}$
(d) $\frac{1}{9}$
The probability distribution of a random variable X is given below :
Table:
| X | 1 | 2 | 3 |
|---|---|---|---|
| P(X) | $\frac{k}{2}$ | $\frac{k}{3}$ | $\frac{k}{6}$ |
(i) Find the value of k.
(ii) Find $P(1 \le X < 3)$.
(iii) Find $E(X)$, the mean of X.
Find the probability that a randomly selected person of the society does Yoga of type A or B but not C.
A problem in Mathematics is given to three students whose chances of solving it are $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}$ respectively. If the events of their solving the problem are independent then the probability that the problem will be solved, is
(a) $\frac{1}{4}$
(b) $\frac{1}{3}$
(c) $\frac...
The random variable $X$ has a probability distribution $P(X)$ of the following form, where 'k' is some real number: $P(X) = \begin{cases} k, \text{ if } x=0 \\ 2k, \text{ if } x=1 \\ 3k, \text{ if } x=2 \\ 0, \text{ otherwise} \end{cases}$ (i) Determine the value of $k$. (ii) Find $P(X < 2)$.
Read the following passage and answer the questions given below: In an Office three employees Jayant, Sonia and Oliver process incoming copies of a certain form. Jayant processes $50%$ of the forms, Sonia processes $20%$ and Oliver the remaining $30%$ of the forms. Jayant has an error rate of $0.06$...
If the sum of numbers obtained on throwing a pair of dice is 9, then the probability that number obtained on one of the dice is 4, is :
(A) $\frac{1}{9}$
(B) $\frac{4}{9}$
(C) $\frac{1}{18}$
(D) $\frac{1}{2}$
Based on the above information, answer the following questions : OR (b) Find the probability that a randomly selected child is left handed given that exactly one of the parents is left handed.
Recent studies suggest that roughly 12% of the world population is left handed. Depending upon the parents, the chances of having a left handed child are as follows : A : When both father and mother are left handed : Chances of left handed child is 24%. B : When father is right handed and mother is ...
Three bags I, II and III contain balls as follows: Bag I: 1 white, 2 black, 3 red balls; Bag II: 2 white, 1 black, 1 red ball; Bag III: 4 white, 5 black, 3 red balls. A bag is chosen at random and two balls are drawn from it. They happen to be one white and one red. What is the probability that they...
(a) Two numbers are selected from first six even natural numbers at random without replacement. If X denotes the greater of two numbers selected, find the probability distribution of X. OR (b) A fair coin and an unbiased die are tossed. Let A be the event, "Head appears on the coin" and B be the eve...
Recent studies suggest that roughly $12%$ of the world population is left handed. Depending upon the parents, the chances of having a left handed child are as follows : A : When both father and mother are left handed : Chances of left handed child is $24%$. B : When father is right handed and mother...
Read the following passage and answer the questions given below : There are ten cards numbered $1$ to $10$ and they are placed in a box and then mixed up thoroughly. Then one card is drawn at random from the box. Based on the above, answer the following questions : (i) What is the probability that t...
(a) A man is known to speak the truth 3 out of 5 times. He throws a pair of different coins and reports that he got a pair of heads. Find the probability that a pair of heads actually occurs.
(b) From a lot of 10 bulbs which includes 2 defectives, a sample of 2 bulbs is drawn at random without replacement. Find the probability distribution of the number of defective bulbs. Hence, find the mean.
X and Y are independent events such that $P(X \cap \bar{Y}) = \frac{2}{5}$ and $P(X) = \frac{3}{5}$. Then P(Y) is equal to :
(a) $\frac{2}{3}$
(b) $\frac{2}{5}$
(c) $\frac{1}{3}$
(d) $\frac{1}{5}$
The probability that A speaks the truth is $\frac{4}{5}$ and that of B speaking the truth is $\frac{3}{4}$. The probability that they contradict each other in stating the same fact is :
(a) $\frac{7}{20}$
(b) $\frac{1}{5}$
(c) $\frac{3}{20}$
(d) $\frac{4}{5}$
Case Study - 1
A building contractor undertakes a job to construct 4 flats on a plot along with parking area. Due to strike the probability of many construction workers not being present for the job is $0.65$. The probability that many are not present and still the work gets completed on time is $0...
Based on the above information, answer the following questions :
(i) What is the probability that all the workers are present for the job ?
(ii) What is the probability that construction will be completed on time ?
(iii) (a) What is the probability that many workers are not present given that the...
Five fair coins are tossed simultaneously. The probability of the events that atleast one head comes up is
(a) $\frac{27}{32}$
(b) $\frac{5}{32}$
(c) $\frac{31}{32}$
(d) $\frac{1}{32}$
Assertion (A) : Two coins are tossed simultaneously. The probability of getting two heads, if it is known that at least one head comes up, is $\frac{1}{3}$.
Reason (R) : Let E and F be two events with a random experiment, then $P(F/E) = \frac{P(E \cap F)}{P(E)}$.
If A and B are two independent events such that $P(A) = \frac{1}{3}$ and $P(B) = \frac{1}{4}$, then $P(B'/A)$ is
(a) $\frac{1}{4}$
(b) $\frac{1}{8}$
(c) $\frac{3}{4}$
(d) $1$
Assertion – Reason Based Questions
In the following questions 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct answer out of the following choices :
(a) Both (A) and (R) are true and (R) is the correct explanation of (A).
(b) Both (A) and (R) are t...
Normal distribution is symmetric about :
(a) variance
(b) standard deviation
(c) mean
(d) covariance
(a) It is given that 2% of screws manufactured by a company are defective. Using Poisson distribution, find the probability that a packet of 100 screws contains no defective screw. (Given : $e^{-2} = 0.14$)
A fair coin is tossed 9 times. Find the probability of getting.
(i) exactly 5 tails;
(ii) at least 5 tails;
(iii) at most 5 tails.
OR
Let X denote the number of hours a person watches T.V. during a randomly selected day. The probability that X can take the values $x_i$, has the following form, where k is some unknown constant.
$P(X = x_i) = \begin{cases} 0.2, & \text{if } x_i = 0 \\ kx_i, & \text{if } x_i = 1 \text{ or } 2 \\ k...
(iii) (b) A student was selected at random and his age was found to be greater than $15$ years. Find the probability that his age is a prime number.
Assertion (A) : Given two independent events A and B such that $ ext{P(A)} = 0\cdot3$ and $ ext{P(B)} = 0\cdot6$, then $ ext{P(A and not B)} = 0\cdot12$.
Reason (R) : For two independent events A and B, $ ext{P(A and B)} = ext{P(A)} \cdot ext{P(B)}$.
A shopkeeper sells three types of flower seeds $A_1$, $A_2$ and $A_3$. They are sold as a mixture where the proportions are $4 : 4 : 2$ respectively. The germination rates of the three types of seeds are 45%, 60% and 35%. Based on the above information, answer the following questions : (i) What is t...
A man is known to speak truth $7$ out of $10$ times. He threw a pair of dice and reports that doublet appeared. Find the probability that it was actually a doublet.
At the start of a cricket match, a coin is tossed and the team winning the toss has the opportunity to choose to bat or bowl. Such a coin is unbiased with equal probabilities of getting head and tail. Based on the above information, answer the following questions : (a) If such a coin is tossed 2 tim...
If such a coin is tossed 2 times, then find the probability distribution of number of tails.
Find the probability of getting at least one head in three tosses of such a coin.
Given two independent events A and B such that $P(A) = 0.3$, $P(B) = 0.6$ and $P(A' \cap B')$ is
(a) $0.9$ (b) $0.18$ (c) $0.28$ (d) $0.1$
Three friends go for coffee. They decide who will pay the bill, by each tossing a coin and then letting the "odd person" pay. There is no odd person if all three tosses produce the same result. If there is no odd person in the first round, they make a second round of tosses and they continue to do s...
Case-Study Based Question: At the start of a cricket match, a coin is tossed and the team winning the toss has the opportunity to choose to bat or bowl. Such a coin is unbiased with equal probabilities of getting head and tail. Based on the above information, answer the following questions : (a) If ...
Calculate the probability that a randomly chosen seed will germinate;
Calculate the probability that the seed is of type A2, given that a randomly chosen seed germinates.
Case Study Problem :
A shopkeeper sells three types of flower seeds $\text{A}_1, \text{A}_2, \text{A}_3$. They are sold in the form of a mixture, where the proportions of these seeds are $4 : 4 : 2$, respectively. The germination rates of the three types of seeds are $45%$, $60%$ and $35%$ respectiv...
A bag contains $3$ red and $4$ white balls. Three balls are drawn at random, one-by-one without replacement from the bag. If the first ball drawn is red in colour, then find the probability that the remaining two balls drawn are also red in colour.
A coin is tossed twice. The following table shows the probability distribution of number of tails:
(a) Find the value of K.
(b) Is the coin tossed biased or unbiased ? Justify your answer.
There are two bags. Bag I contains 1 red and 3 white balls, and Bag II contains 3 red and 5 white balls. A bag is selected at random and a ball is drawn from it. Find the probability that the ball so drawn is red in colour.
Three friends A, B and C got their photograph clicked. Find the probability that B is standing at the central position, given that A is standing at the left corner.
Let $X$ be a random variable which assumes values $x_1, x_2, x_3, x_4$ such that $2P(X=x_1) = 3P(X=x_2) = P(X=x_3) = 5P(X=x_4)$. Find the probability distribution of $X$.
Three persons A, B and C apply for a job of manager in a private company. Chances of their selection are in the ratio $1 : 2 : 4$. The probability that A, B and C can introduce changes to increase the profits of a company are $0.8$, $0.5$ and $0.3$ respectively. If increase in the profit does not ta...
Two cards are drawn at random without replacement from a well-shuffled deck of 52 playing cards. Find the probability of getting both cards of the same colour.
A bag contains 2 white, 2 red and 3 blue balls. Two balls are drawn at random from the bag one-by-one with replacement. Find the probability distribution of the number of white balls.
In a class, 5% of the boys and 10% of the girls have an IQ more than 150. In this class, 60% of the students are boys. One student is selected at random from the class. Based on the above, (a) Find the probability that the selected student has an IQ more than 150. (b) If it is given that the selecte...