Answer:
135/512
Step-by-step 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
