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STATEMENT

In the card game bridge, the 52 cards are dealt out equally to 4 players—called East, West, North, and South. If North and South have a total of 8 spades among them, what is the probability that East has 3 of the remaining 5 spades?

HINT: use conditional probability



SOLUTION Since East and West have half the cards now, so the new sample space for conditional probability = $26$ out of which $5$ are spades and hence the conditional probability of East getting 3 spades is:

$$\frac{13 \choose 3} {26 \choose 5} = 0.1$$

Correct answer is $0.34$ instead.

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ArnuldOnData
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1 Answers1

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Your initial analysis is correct, there are 26 cards distributed between East and West, exactly 5 of which are spade. Your choice of binomial coefficients to represent the asked for proabbility, however, makes no sense to me.

East has 13 of those 26 cards, so there are $26 \choose 13$ possible hands East could have. How many contain exactly 3 spades? For that to be true, East must have a hand of 10 non-spades, which comes from 21 non-spades among the cards under consideration. In addition, East must have 3 spades, which comes from 5 spades among the cards under consideration. That means there are ${21 \choose 10}{5 \choose 3}$ 'good' hands for East.

Since all hands are assumed to be equiprobable, the desired probability is

$$p=\frac{{21 \choose 10}{5 \choose 3}}{26 \choose 13} \approx 0.34.$$

Ingix
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