Question

Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. What is the probability that

1. all the five cards are spades?
2. only 3 cards are spades?
3. none is a spade?

Answer 1

Step 1: Define the parameters

Let X represent the number of spade cards among the five cards drawn. Since the drawing of a card is with replacement, the trials are Bernoulli trials.

In a well-shuffled deck of 52 cards, there are 13 spade cards.

⇒ `p = 13/52 = 1/4`

∴ `q = 1 - 1/4 = 3/4`

X has a binomial distribution with n = 5 and p = `1/4`

Step 2: Probability of all 5 cards being spades:

To find the probability that all 5 cards drawn are spades, we need to calculate P(X = 5):

`P(X = x) =  ^nC_xq^n-^x p^x`, where x = 0, 1, ... n

= `"^5C_x (3/4)^(5-x) (1/4)^x`

P (all five cards are spades) = P(X = 5)

= `"^5C_5 (3/4)^0 · (1/4)^5`

= `1 · 1/1024` 

= `1/1024`

Step 3: Probability of exactly 3 cards being spades

P (only 3 cards are spades) = P(X = 3)

= `"^5C_3 · (3/4)^2 · (1/4)^3`

= `10 · 9/16 · 1/64`

= `45/512`

Step 4: Probability of none being spades

P (none is a spade) = P(X = 0)

= `"^5C_0 · (3/4)^5 · (1/4)^0`

= `1 · 243/1024`

= `243/1024`

Summary of Results:

1. Probability that all 5 cards are spades: `1/1024`
2. Probability that exactly 3 cards are spades: `45/512`
3. Probability that none is a spade: `243/1024`