 Welcome to the session. I am Deepika and I am going to help you solve the following question. The question says, suppose x has a binomial distribution v6 1 over 2 show that x is equal to 3 is the most likely outcome. Now we know that the probability of x successes is given by ncx q raise to power n minus x into p raise to power x, where x is from 0 to n q is equal to 1 minus p. Also a binomial distribution with n but not the trials and probability of success in each trial as p is denoted by b and p. So this is a key idea behind our question. We will take the help of this key idea to solve the above question. So let's start the solution. Now we are given x has a binomial distribution v6 1 over 2 that is x has a binomial distribution v6 1 over 2. So according to our key idea this implies n is equal to 6. p is equal to 1 over 2 therefore q which is equal to 1 minus p is equal to 1 minus 1 over 2 and this is again equal to 1 over 2. Now we have to show that x is equal to 3 is the most likely outcome. That is we have to show that if we take x is equal to 3 we get the maximum value as compared to x is equal to 0, 1, 2, 4, 5, 6. Since n is equal to 6 p is equal to 1 over 2 and q is also equal to 1 over 2 we have probability of x successes is equal to 6 c x into 1 over 2 raise to power 6 minus x into 1 over 2 raise to power x and this is equal to 6 c x 1 over 2 raise to power 6. Now probability of 0 success is equal to 6 c 0 into 1 over 2 raise to power 6. Again probability of 1 success is equal to 6 c 1 into 1 over 2 raise to power 6. Similarly probability of 2 successes is equal to 6 c 2 into 1 over 2 raise to power 6 and probability of 3 successes is equal to 6 c 3 into 1 over 2 raise to power 6. Now we have to find the probabilities from 0 to 6 successes. Now here we observe that we will always get 1 over 2 raise to power 6 in all the expressions. Now we know that 6 c 0 is equal to 6 c 6 which is equal to 1 because n c r is equal to n c n minus r therefore 6 c 0 is equal to 6 c 6. Similarly 6 c 1 is equal to 6 c 5 which is equal to 6 because n c r is equal to n c n minus r again 6 c 2 is equal to 6 c 4 which is equal to 15. Now 6 c 3 is equal to 20 thus here we observe that 6 c 3 has a maximum value as compared to 6 c 0, 6 c 1, 6 c 2, 6 c 4, 6 c 5 and 6 c 6. So probability of 3 successes which is equal to 6 c 3 into 1 over 2 raise to power 6 has value greater than other probabilities x is equal to 0, 1, 2, 4, 5, 6 hence we can say x is equal to 3 is the most likely outcome. We have proved that x is equal to 3 is the most likely outcome. So this completes our session. I hope the solution is clear to you. Bye and have a nice day.