Question

PROBLEM 1. There are 12 cards in a standard deck of cards with faces on them, namely the 4 Jacks, the 4 Queens, and the 4 Kings. Assume below that the deck is well shuffled.

(a). If you deal 5 cards without replacement, what is the probability of getting no face cards?

(b). If you deal 5 cards WITH replacement (each time replacing the previous card and shuffling before dealing the next card), what is the probability of getting no face cards?

REMARK: You should know which of (a) and (b) will have the larger answer even before doing the arithmetic.

(c). What is the expected number of face cards dealt in part (a)?

(d) What is the expected number of face cards dealt in part (b)?

Answer 1

There are 12 face cards and 40 non face cards in the deck. We draw 5 cards without replacement from the deck.

The Total number of ways to draw 5 cards from a deck of 52 cards is

Let X=x be the number of face cards in 5 cards that are drawn without replacement.

The number of ways of drawing X facecards from 12 face cards is

The remaing 5-x cards out of 5 drawn are non face cards. They are drawn out of 40 non face cards. The total niumber of ways of drawing 40x non face vards out of 40 non face cards =

The probability of drawing X=x face cards is

The above distribution is also called Hyper geometric distribution

The probability of getting no face cards is probability of X=0

b) There are 12 face cards out of 52 total cards. The probability of choosing a face card is

The probaility of choosing a non face card is

these probabilities remain the same for 5 draws as we are replacing the card after each draw

Let X=x be the number of face cards drawn out of 5 draws. the joint probability of getting X=x face cards and 5-x non face cards is

There are ways of getting X=x face cards out of 5 draws.

the probability of getting X=x face cards is

This is a Binomial distribution

the probability of getting no face cards is probability of X=0

We know that in b the probability of drawing a non face card remains constant at 40/52 = 0.7692 through out the 5 draws (as the cards are being replaced).

In a) the probability of drawing a non face card in the frist draw is 40/52 = 0.7692, but in the second draw, the conditional probability of drawing a non face card given that it is a non face card in the first draw decreases to 39/51=0.7647 It reduces further as we get to the 5th draw. Hence we can say that the probability of drawing no face cards is higher in b compared to a

c) the expected number of face cards is expected value of X.

d) The expected value of X in part b is