Question

1A. We draw five cards at random with replacement, keeping track of whether or not each card in the sequence is a club. Find the probability of the following sequence: Club, Not Club, Club, Not Club, Not Club, i.e. CNCNN.

B. Notice that all sequences of 5 cards with exactly two clubs have the same probability. Based on this fact and your result in a), find the probability of getting exactly 2 clubs in a sequence of 5 cards drawn at random with replacement from a 52 card deck.

Answer 1

135/512

Explanation:

In a deck of card, there are 13clubs.

A deck of card contains 52cards

Total clubs = 13cards

Probability of selecting a club and replacing is expressed as P(C) = total number of clubs/total card

P(C) = 13/52 = 1/4

q = 1-p

q = 1-1/4

q =3/4

Let X be probability function used is a binomial distribution formula expressed as:

Since 5cards is drawn at random, n = 5

P(X = x) = nCx q^(n-x)p^x

P(X = x) = 5Cx 3/4^(5-x)•(1/4)^x

The probability of the sequence is expressed as:

P(X = x) = 5Cx 3/4^(5-x)•(1/4)^x

x is the number of clubs present

b) To find the probability of getting exactly 2 clubs in a sequence of 5 cards drawn at random with replacement from a 52 card deck, we will substitute x = 2 into the probability function in (a) a shown;

P(X = 2) = 5C2•3/4^(5-2)•(1/4)^2

P(X = 2) = 5C2•3/4^(3)•(1/4)^2

P(X = 2) = 5C2•27/64•(1/16)

P(X = 2) = 5C2•27/1024

P(X = 2) = 5!/(5-2)!2!•27/1024

P(X = 2) = 5!/3!2!•9/1024

P(X = 2) = 5×4×3!/3!2•27/1024

P(X = 2) = 5×4/2•27/1024

P(X = 2) = 10•27/1024

P(X = 2) = 135/512

Hence the probability of getting exactly 2 clubs in a sequence of 5 cards drawn at random with replacement from a 52 card deck is 135/512