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Chapter 8: Problem 451

A hand of five cards is to be dealt at random and without replacement from an ordinary deck of 52 playing cards. Find the conditional probability of an all spade hand given that there will be at least 4 spades in the hand.

Short Answer

Expert verified
The conditional probability of an all spade hand given that there are at least 4 spades in the hand is: \(P(all \ spades \ |\ at \ least \ 4 \ spades) = \frac{C(13, 5)}{C(13, 4) \times C(39, 1) + C(13, 5)}\)

Step by step solution

01

Identify the Combinations Needed to Calculate the Probabilities

There are three combinations we need to find: 1. The total number of combinations of 5 cards out of 52 (this will be the denominator of the probabilities we calculate). 2. The number of combinations of getting 4 spades and 1 non-spade. 3. The number of combinations of getting 5 spades (which is equal to the number of all spade hands).
02

Calculate Total Number of Combinations of 5 Cards out of 52

To find the total number of combinations of 5 cards out of 52, we use the following formula: \(C(n, r) = \frac{n!}{r!(n - r)!}\) where n is the total number of cards, and r is the number of cards selected. Plugging in the values, we get: \(C(52, 5) = \frac{52!}{5!(52 - 5)!}\)
03

Calculate Combinations for 4 Spades and 1 Non-Spade

To find the number of combinations for 4 spades and 1 non-spade, we need to find the number of ways to pick 4 spades from the 13 in the deck and 1 non-spade from the remaining 39 cards. This can be done using the combination formula twice: \(C(13, 4) = \frac{13!}{4!(13 - 4)!}\) (number of ways to pick 4 spades) \(C(39, 1) = \frac{39!}{1!(39 - 1)!}\) (number of ways to pick 1 non-spade) Now we multiply these two values to find the total combinations of 4 spades and 1 non-spade: \(C(13, 4) \times C(39, 1)\)
04

Calculate Combinations for an All Spade Hand

To find the number of combinations for an all spade hand, we simply need to find the number of ways to pick 5 spades from the 13 in the deck: \(C(13, 5) = \frac{13!}{5!(13 - 5)!}\)
05

Calculate Probability of at Least 4 Spades in the Hand

To find the probability of at least 4 spades in the hand, we need to add the probabilities of getting exactly 4 spades and getting 5 spades: \(P(at \ least \ 4 \ spades) = \frac{C(13, 4) \times C(39, 1) + C(13, 5)}{C(52, 5)}\)
06

Calculate the Conditional Probability of an All Spade Hand Given at Least 4 Spades

Finally, we can find the conditional probability by dividing the probability of an all spade hand by the probability of at least 4 spades: \(P(all \ spades \ |\ at \ least \ 4 \ spades) = \frac{P(all \ spades)}{P(at \ least \ 4 \ spades)} = \frac{C(13, 5)}{C(13, 4) \times C(39, 1) + C(13, 5)}\) Calculate each of the combinations and probabilities to find the conditional probability.

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