 Hello and welcome to the session. The given question says, two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the probability distribution of number of jacks. Let's start with the solution. So in a pack of well shuffled card there are 52 cards out of which jack cards are four. Therefore no jack card is given by 52-4 which is equal to 48 cards. So in a well shuffled pack of 52 cards 48 cards are non-jack cards and four are jack cards. Let x denote the number of jacks. This implies x can take the values 0, 1 or 2 since two cards are drawn. Firstly let's find the probability of x is equal to 0 that is probability that no jack card is drawn. Now there are 48 cards which are not jack cards. Therefore we have 48 C2 which are the number of events favorable that no jack card is drawn divided by the total number of possible events which is 52 C2. So this is equal to 48 factorial divided by 2 factorial divided by 48-2 that is 46 factorial whole divided by factorial 52 divided by factorial 2 into factorial 50. So this is equal to 48 into 47 into factorial 46 divided by factorial 2 into factorial 46 into factorial 50 divided by 52 into 51 into factorial 50. Now on simplifying it we get 48 into 47 divided by 52 into 51 and on simplifying this further we get 188 divided by 221. Now let's find probability when x is equal to 1 that is probability that one card is jack other is a non-jack card. Now there are four jack cards so we have four C1 into and 48 non-jack cards so we have 48 C1 whole divided by 52 C2. So this is further equal to factorial 4 divided by factorial 1 into factorial 3 into factorial 48 divided by factorial 1 into factorial 47 divided by factorial 52 whole divided by factorial 2 into factorial 50. So this is further equal to 4 into 48 divided by 52 into 51 into 2 and this is equal to 32 divided by 221. Now let's find probability when x is equal to 2 that is probability that both jack cards are drawn. This is equal to 4 C2 divided by 52 C2 since there are four jack cards and out of these four we have to draw two. So we have 4 C2 as the probability. Now this is equal to factorial 4 divided by factorial 2 into factorial 2 into factorial 2 into factorial sorry this is 2 into factorial 50 divided by factorial 52 which is further equal to 4 into factorial 2 into 3 since factorial 4 is equal to 4 into 3 into factorial 2 divided by factorial 2 into factorial 2 into factorial 2 into factorial 50 divided by 52 into 51 into factorial 50. So this one simplifying further gives equal to 4 into 3 divided by 52 into 51 now 4 into 13 gives 52 and 3 into 17 gives 51 this is equal to 1 divided by 221 therefore we have probability at x is equal to 0 equal to 188 divided by 221 probability when x is equal to 1 is 32 divided by 221 and probability when x is equal to 2 is equal to 1 divided by 221 thus let us make a probability distribution table in the first row we shall write the values of x and in the second row we shall write the values of px so when x is 0 1 and 2 the value of px are 188 divided by 221 32 divided by 221 and 1 divided by 221 so this completes the session buy and take care.