Question

Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. What is the probability that
(i) all the five cards are spades?
(ii) only 3 cards are spades?
(iii) none is a spade?

Answer 1

Answer:

Step-by-step explanation:

Hope this helps

There are 52 cards in a deck, out of which 13 are spades.

The experiment is to draw (n = 5) cards w replacement. It is a case of Bernoulli trials as it satisfies the conditions (i) finite number of trials, (ii) independent trials, (iii) there is a definite outcome and (iv) the probability of success does not change for each trial.

P ( a spade is drawn ) = p = 1352=141352=14

P ( a card other than spade is drawn ) = q = 1 - p = 1−1352=341−1352=34

Since X has a bionomial distribution, the probability of x success in n-Bernoulli trials, P(X=x)=nCx.px.qn–xP(X=x)=nCx.px.qn–x where x=0,1,2,...,nx=0,1,2,...,n and (q=1–p)(q=1–p)

Here n=5,p=14,n=5,p=14,q=34q=34.

(i) Probability that if 5 cards are drawn all of them are spades:

P(X=5)=5C5.145.345–5=P(X=5)=5C5.145.345–5= 145=11024145=11024

(ii) Probability that only 3 cards are spades = P (3 spades and 2 non spades):

P (X = 3) = 5C31435C3143××345345 = 606164606164××916=901024916=901024

(iii) Probability that there are no spades = P (0 spades):

P (X = 0) = (ii) Probability that only 3 cards are spades = P (3 spades and 2 non spades):

P(X=0)=5C0.140.345–0=(34)5=2431024