Question

We draw the top 5 cards from a well-shuffled standard 52-card deck. Find the probability that:

a) The first two cards are Kings and the remaining three cards are Queens. (3 marks)

b) The 5 cards include exactly 2 Kings and 3 Queens. (5 marks)

c) The 5 cards include exactly 2 Kings, or exactly 1 Queen, or both. (7 marks)

Answer 1

Number of ways of selecting 5 cards out of 52 cards is

$\binom{52}{5}=2598960$

(a)

There are 4 kings and 4 queens so the number of ways of selecting 2 kings and 3 queens is

The probability that the first two cards are Kings and the remaining three cards are Queens is

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Number of ways of selecting 2 kings and three other cards is

So,

The probability that the first two cards are Kings and the remaining three cards are Queens is

Answer: 0.00023

(b)

There are 4 kings and 4 queens so the number of ways of selecting 2 kings and 3 queens is

The probability that the first two cards are Kings and the remaining three cards are Queens is

Answer: 0.00000923

(c)

Number of ways of selecting 2 kings and three other cards is

So,

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Number of ways of selecting 1 queen and four other cards is

So,

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Number of ways of selecting 2 kings and 1 queen and 2 other cards is

So,

The probability that the 5 cards include exactly 2 Kings, or exactly 1 Queen, or both is

Answer: 0.3307