 Hello and welcome to the session. I am Deepika here. Let's discuss the question which says Five cards are drawn Successively with replacement from a well shuffled deck of 52 cards What is the probability that all the five cards are spades? Only three cards are spades And one is a spade Now we know that the trials of a random experiment are called Bernoulli trials If they are finite in number, 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, the probability of x successes is given by ncx into q raised to power n minus x into p raised to power x where x is from 0 to n and q is equal to 1 minus p So this is a 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 According to the question, five cards are drawn successively with replacement from a well shuffled deck of 52 cards Since cards are drawn with replacement, number of trials is finite That is five The trials are Bernoulli trials Or we can say the drawing of cards with replacement Bernoulli trials Now we have to find the probability that all the five cards are spades Only three cards are spades and none is a spade Let x denote the number of spades Five cards drawn x has a binomial distribution With n is equal to five and p which is a probability of success is equal to 13 over 52 Because there are 13 spades in a deck of 52 cards So p is equal to 1 over 4 Therefore q which is given by 1 minus p is equal to 1 minus 1 over 4 And this is equal to 3 over 4 Now according to our key idea We have probability of x success is equal to 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 Now here we have n is equal to 5 p is equal to 1 over 4 And q is equal to 3 over 4 Therefore probability of x success is equal to 5cx into 3 over 4 raise to power 5 minus x into 1 over 4 Raise to power x Now in part one we have to find the probability that all the five cards are spades This means we have to find the probability of five successes This is equal to 5c5 into 3 over 4 raise to power 5 minus 5 Into 1 over 4 raise to power 5 And this is again equal to 1 into 3 over 4 raise to power 0 Which is again 1 into 1 over 4 raise to power 5 Which is 1 over 1024 And this is again equal to 1 over 1024 Hence the probability that all the five cards are spades is 1 over 1024 So this is the answer for part one Now in part two we have to find the probability that only three cards are spades That is we have to find the probability of three successes So this is equal to 5c3 into 3 over 4 raise to power 5 minus 3 into 1 over 4 raise to power 3 Now this is equal to 5c3 which is 5 factorial over 3 factorial into 2 factorial into 3 over 4 Raise to power 2 which is 9 over 16 into 1 over 4 cube which is 1 over 64 And this is again equal to 10 into 9 over 16 into 1 over 64 Which is equal to 90 over 1024 And this is again equal to 45 over 512 Hence the probability that only three cards are spades is 45 over 512 So this is the answer for part two Now in part three we have to find the probability that none is a spade That is we have to find the probability of zero successes And this is equal to 5c0 into 3 over 4 raise to power 5 minus 0 into 1 over 4 raise to power 0 Now 5c0 is 1 and 3 over 4 raise to power 5 is 243 over 1024 And 1 over 4 raise to power 0 is 1 So this is equal to 1 into 243 over 1024 into 1 And this is again equal to 243 over 1024 Hence the probability that none of the card is a spade is 243 over 1024 So this is the answer for part three So this completes our session I hope the solution is clear to you Bye and have a nice day