Question

7) Suppose that you select 2 cards without replacement from a standard deck of 52 playing cards.

a) If the first card that you select is not a heart, what is the probability that the second card that you select is a heart?

b) If the first card that you select is a six, what is the probability that the second card that you select is a diamond?

8) Suppose that a teacher is going to assign a book report to her class of 22 students. Each student must select one book from an approved reading list and once a book is selected by a student, no other student may select the same book. The reading list consists of a total of 39 books of which 15 are considered classic fiction and the other 24 are considered modern fiction. Assuming that the order in which the books are selected doesn’t matter:

a) In how many ways can the books be selected so that all of the classic fiction books are picked?

b) In how many ways can the books be assigned so that there is at least one classic fiction book that is not picked and at least one modern fiction book that is not picked.

Answer 1

7)

Total number of ways of selecting two cards where te first card is not heart = 39C1 * 51C1., because for first card it can be one of diamond, spade and club, i.e any one of 13*3=39 cards, and for the 2nd there is no restriction, so anyone of (52-1) = 51 as the one is seleccted can't come as it is without replacement..

Out of them the number ways of selecting two cards where first card is not heart and 2nd card is heart = 39C1*13C1

As 39 spade ,club and damond are disjoint of 13 hearts.

So the conditional probability of selecting the second card as heart given that the first card is not heart

= (39C1*13C1)/(39C1*51C1)= 0.25

(b)

Similiarly the number of ways of selecting 2 cards where first one six = 4C1 * 51C1= 51*4, beacuse there are total 13 denomination, each of 4 cards, so first one must be chosen from the 4 cards with denomination 6 and second one will be chosen from the remaining 51 cards.

Out of them number of ways of selecting two cards where 1st one is six and 2nd one is diamond

= 3C1 * 13C1 + 1C1 * 12C1 = 39+12 = 51., because out of 4 cards with 6 denominations ther are one spade, one club, one heart, one diamond. As the two cards are chosen without replacement, so if the first card chosen is diamond six then 2nd card must be chosen from the remaining (13-1)=12 diamonds. the total number of ways of choosing it = 1C1*12C1 = 12 . For other cars eif the first one is chosen as six of either spade, heart or club then the 2nd one will be chosen from the whole set of 13 diamonds. The number of ways i = 3C1 * 13C1=39. So total number of ways = 39+12 =51.

So the required probability = 51/51*4 = 1/4 = 0.25

8)

Total number of books is 39

Classic fiction book is 15

Modern fiction book is 24

Number of students is 22

number of ways to select the book by student is C(39,22)

number of ways all the books are classic fiction is C(15,15)

number of ways all the books are modern fiction is C(24,7)

a) many ways can the books be selected so that all of the classic fiction books are picked is

(15c15)*(24c7) =

b) many ways can the books be assigned so that there is at least one modern fiction book that is not picked and at least one classic fiction book that is not picked.

number of ways there is at least one classic fiction book that is not picked is C(15,0)*C(24,24)

number of ways there is at least one modern fiction book that is not picked is C(24,0)*C(15,15)