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$_2$ and 40% of the population was given Vaccine A$_3$. The survey also stated that the probabilities that Vaccines A$_1$, A$_2$ and A$_3$ would protect against the infection were 60%, 55% and 50% respectively. Based on the above information, answer the following questions : Find the probability that :