you are dealt 13-card bridge hand from a 52-card bridge deck.
what is the probability that you will be dealt exactly 5 hearts?
what is the probability that you will dealt at list 2 hearts?
what is the probability that you will receive 5 hearts, 4 clubs, 2 diamonds, and 1 spade?
To answer these probability questions, we need to calculate the number of favorable outcomes (hands that satisfy the given condition) and the total number of possible outcomes (all possible hands).
Total possible outcomes:
In bridge, each player is dealt 13 cards from a 52-card deck. Hence, the total number of possible outcomes is given by the combination formula:
C(52, 13) = 52! / (13!(52-13)!) = 635,013,559,600.
1. Probability of being dealt exactly 5 hearts:
To calculate this, we need to determine the number of favorable outcomes, which is the number of ways to choose 5 hearts from the 13 available hearts, multiplied by the number of ways to choose the remaining 8 cards (non-hearts) from the remaining 39 cards (52 cards - 13 hearts).
Favorable outcomes = C(13, 5) * C(39, 8) = (13! / (5!(13-5)!)) * (39! / (8!(39-8)!) = 1,101,770,576,480.
Therefore, the probability of being dealt exactly 5 hearts is:
P(5 hearts) = Favorable outcomes / Total outcomes = 1,101,770,576,480 / 635,013,559,600 ≈ 0.0017 or 0.17%.
2. Probability of being dealt at least 2 hearts:
To calculate this, we need to calculate the probability of the complementary event (not being dealt at least 2 hearts) and subtract it from 1.
Favorable outcomes for not being dealt at least 2 hearts = C(13, 0) * C(39, 13) + C(13, 1) * C(39, 12).
= (13! / (0!(13-0)!)) * (39! / (13!(39-13)!)) + (13! / (1!(13-1)!)) * (39! / (12!(39-12)!))
= 508,313,959,260 + 1,270,784,824,075
= 1,779,098,783,335.
Therefore, the probability of being dealt at least 2 hearts is:
P(at least 2 hearts) = 1 - (Favorable outcomes for not being dealt at least 2 hearts / Total outcomes)
= 1 - (1,779,098,783,335 / 635,013,559,600) ≈ 0.7192 or 71.92%.
3. Probability of receiving 5 hearts, 4 clubs, 2 diamonds, and 1 spade:
To calculate this, we need to determine the number of favorable outcomes, which is the number of ways to choose 5 hearts from the 13 available hearts, multiplied by the number of ways to choose 4 clubs from the 13 available clubs, multiplied by the number of ways to choose 2 diamonds from the 13 available diamonds, multiplied by the number of ways to choose 1 spade from the 13 available spades. Then, multiply this by the number of ways to choose the remaining 1 card from the remaining 31 cards (52 cards - 13 hearts - 13 clubs - 13 diamonds - 1 spade).
Favorable outcomes = C(13, 5) * C(13, 4) * C(13, 2) * C(13, 1) * C(31, 1)
= (13! / (5!(13-5)!)) * (13! / (4!(13-4)!)) * (13! / (2!(13-2)!)) * (13! / (1!(13-1)!)) * (31! / (1!(31-1)!)).
= 488,551,536,704,000.
Therefore, the probability of receiving 5 hearts, 4 clubs, 2 diamonds, and 1 spade is:
P(5 hearts, 4 clubs, 2 diamonds, 1 spade) = Favorable outcomes / Total outcomes = 488,551,536,704,000 / 635,013,559,600 ≈ 0.7693 or 76.93%.