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Chapter 1: Problem 66

Compute the probability of being dealt at random and without replacement a 13 -card bridge hand consisting of: (a) 6 spades, 4 hearts, 2 diamonds, and 1 club; (b) 13 cards of the same suit.

Short Answer

Expert verified
The probability for scenario (a) is given by the expression [C(13,6)*C(13,4)*C(13,2)*C(13,1)] / C(52,13). The probability for scenario (b) is given by the expression [4*C(13,13)] / C(52,13). Note that the actual values would require further computation using the combination formula.

Step by step solution

01

Calculate possibility for scenario (a)

Compute the probability of getting 6 spades out of 13, 4 hearts out of 13, 2 diamonds out of 13, and 1 club out of 13. Apply the combination formula \(C(n,k) = n! / [(n-k)!k!]\), where n represents the sample space and k is the event we are finding the probability of. In this case: \(P(A) = [C(13,6]*C(13,4)*C(13,2)*C(13,1)] / C(52, 13)\)
02

Calculate possibility for scenario (b)

Compute the probability of getting 13 cards of the same suit. This essentially amounts to selecting 13 cards from any given suit. So, 'n' is 13 in this case and 'k' is also 13. Apply the combination formula: \(P(B) = [4*C(13,13)] / C(52,13)\), as there are 4 suits from which we can draw the 13 cards.
03

Calculate the combinations

Calculate separately all of the involved combinations. Find C(13,6), C(13,4), C(13,2), C(13,1), C(13,13) and C(52,13).

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