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Question
Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. What is the probability that
- all the five cards are spades?
- only 3 cards are spades?
- none is a spade?
Solution
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:
- Probability that all 5 cards are spades: `1/1024`
- Probability that exactly 3 cards are spades: `45/512`
- Probability that none is a spade: `243/1024`
Notes
Students should refer to the answer according to their question and preferred marks.
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