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Chapter 1: Problem 4

From a well-shuffled deck of ordinary playing cards, four cards are turned over one at a time without replacement. What is the probability that the spades and red cards alternate?

Short Answer

Expert verified
The probability that the spades and red cards alternate is \(2 \times \frac{{2 \times {13 \choose 2} \times {26 \choose 2}}}{{52 \choose 4}}\)

Step by step solution

01

Compute the probability of first situation

Consider the situation where you start with a spade. The total ways you can choose 2 spades and 2 red cards are given by \({13 \choose 2} \times {26 \choose 2}\). To alternate starting with a spade, the possibilities for the orders are SRSR or SR. The probability, therefore, is \(\frac{{2 \times {13 \choose 2} \times {26 \choose 2}}}{{52 \choose 4}}\)
02

Compute the probability of second situation

Now consider the situation where you start with a red card. Again, the total ways you can choose 2 spades and 2 red cards are given by \({13 \choose 2} \times {26 \choose 2}\). To alternate starting with a red card, the possibilities for the orders are RSRS or RS. The probability, therefore, is \(\frac{{2 \times {13 \choose 2} \times {26 \choose 2}}}{{52 \choose 4}}\)
03

Sum the probabilities

The final step is to add up the probabilities calculated in the first two steps. So the total probability is \(2 \times \frac{{2 \times {13 \choose 2} \times {26 \choose 2}}}{{52 \choose 4}}\)

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