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Chapter 3: Problem 10

Three cards are randomly selected, without replacement, from an ordinary deck of 52 playing cards. Compute the conditional probability that the first card selected is a spade given that the second and third cards are spades.

Short Answer

Expert verified
The conditional probability that the first card selected is a spade, given that the second and third cards are spades, is \(\frac{11}{50}\).

Step by step solution

01

Identify the events

Let A be the event that the first card selected is a spade, and let B be the event that the second and third cards are spades.
02

Find the probability of B

There are 13 spades in a deck of 52 cards. The probability of drawing two spades in a row (the event B) is: \(P(B) = \frac{13}{52} \times \frac{12}{51}\)
03

Find the probability of A and B

The event A and B means that all three cards are spades. The probability of drawing three spades in a row is: \(P(A \cap B) = \frac{13}{52} \times \frac{12}{51} \times \frac{11}{50}\)
04

Apply the formula for conditional probability

Now we will apply the formula for conditional probability, P(A|B) = P(A and B) / P(B). Thus: \(P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{13}{52} \times \frac{12}{51} \times \frac{11}{50}}{\frac{13}{52} \times \frac{12}{51}}\)
05

Cancel the common terms and calculate the result

We can cancel out common terms in the numerator and denominator: \(P(A|B) = \frac{\frac{11}{50}}{1}\) So the conditional probability that the first card selected is a spade, given that the second and third cards are spades, is \(\frac{11}{50}\).

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