 Hello and welcome to the session, I am Deepika here. Let's discuss the question which says A person buys a lottery ticket in 50 lotteries in each of which his chance of winning a prize is 1 over 100 What is the probability that he will win a prize at least once, exactly once, at least twice Now we know that trials of a random experiment are called Bernoulli trials If they are finite in numbers, they are independent and each trial has exactly two outcomes Success or failure and the probability of success remains the same in each trial Again, probability of x successes is given by ncx into q raise to power n minus x into p raise to power x where x is from 0 to n and q is equal to 1 minus p So this is the key idea behind the question We will take the help of this key idea to solve the above question So let's start the solution So according to the given question, a person buys a lottery ticket In 50 lotteries, each of which his chance of winning a prize is 1 over 100 So event of buying a lottery ticket in 50 lotteries is independent of each other Also number of trials is finite that is 50 Therefore the trials are Bernoulli trials Let x denote the number of chances when a person wins a prize So clearly as the binomial distribution with n is equal to 50 and p is equal to 1 over 100 That is probability of the success is 1 over 100 Now according to our key idea, probability of x successes is given by ncx into q raise to power n minus x into p raise to power x where x is from 0 to n and q is equal to 1 minus p Here we have is equal to 50, p is equal to 1 over 100 So q is equal to 1 minus 1 over 100 which is again equal to 99 over 100 Therefore probability of x successes is equal to 50cx into 99 over 100 raise to power 50 minus x into 1 over 100 Raise to power x, we have to find the probability that the person will win a prize at least once So probability of at least once is given by probability of x greater than equal to 1 Now probability of x greater than equal to 1 can be written as 1 minus probability of x equal to 0 probability of at least once is equal to 1 minus probability of winning not even once That is probability of at least once is equal to 1 minus probability of 0 success So this is equal to 1 minus 50c0 into 99 over 100 raise to power 50 minus 0 into 1 over 100 raise to power 0 Now 50c0 is 1 also 1 over 100 raise to power 0 is 1 So probability of at least once is equal to 1 minus 99 over 100 raise to power 50 So this is the answer for part A In part B we have to find the probability that he will win a prize exactly once So probability of exactly once is given by probability of x equal to 1 And this is equal to 50c1 into 99 over 100 raise to power 50 minus 1 into 1 over 100 raise to power 1 Now this is again equal to 50c1 which is 50 into 99 over 100 raise to power 49 into 1 over 100 And this is again equal to 1 over 2 into 99 over 100 raise to power 49 So this is the answer for part B Now in part C we have to find the probability that he will win a prize at least twice So probability of winning at least twice is given by probability of x greater than equal to 2 Now we know that probability of winning at least twice is equal to 1 minus probability of winning less than twice So probability of x greater than equal to 2 is equal to 1 minus probability of x less than 2 That is probability of x equal to 0 and 1 And this is equal to 1 minus 50c0 into 99 over 100 raise to power 50 minus 0 into 1 over 100 raise to power 0 Plus 50c1 into 99 over 100 raise to power 50 minus 1 into 1 over 100 raise to power 1 This is again equal to 1 minus 50c0 which is 1 into 99 over 100 raise to power 50 into 1 because 1 over 100 raise to power 0 is 1 plus 50c1 which is 50 into 99 over 100 raise to power 49 into 1 over 100 And this is again equal to 1 minus 99 over 100 raise to power 50 plus 1 over 2 into 99 over 100 raise to power 49 Now again this can be written as 1 minus 99 over 100 raise to power 49 into 99 over 100 plus 1 over 2 And this is equal to 1 minus 99 over 100 raise to power 49 into 99 plus 50 over 100 And this is again equal to 1 minus 149 over 100 into 99 over 100 raise to power 49 So this is the answer for part c This completes our session I hope the solution is clear to you Bye and have a nice day