(a) What are the chances of being dealt one pair or a full house?
(b) What are the chances of being dealt a flush or better?
(c) What are the chances of being dealt a hand that will lose to a straight?
To determine the chances of being dealt certain poker hands, we need to calculate the probabilities. In poker, a standard deck of 52 cards is used.
(a) To calculate the chances of being dealt one pair or a full house, we need to consider the number of possible combinations for these hands.
One pair: Each card can be any of the 13 ranks, and there are 4 possible suits for each rank. Therefore, the number of possible one pair combinations is:
13 ranks * (4 suits choose 2) * (12 ranks choose 3) * (4 suits choose 1) * (4 suits choose 1) = 1,098,240
Full house: There are 13 ways to choose the rank for the three cards in the full house. Similarly, there are 4 suits for each of these ranks. Thus, the number of full house combinations is:
13 ranks choose 1 * (4 suits choose 3) * 12 ranks choose 1 * (4 suits choose 2) = 3744
To calculate the probability, we divide the number of combinations for one pair or full house by the total number of possible poker hands:
Probability = (1,098,240 + 3,744) / (52 cards choose 5) = 0.228%.
Therefore, the chances of being dealt one pair or a full house are approximately 0.228%.
(b) To calculate the chances of being dealt a flush or better, we need to consider the combinations for flushes, straights, straight flushes, and royal flushes.
Flush: There are 4 ways to choose a suit, and each suit has 13 cards. Thus, the number of flush combinations is:
4 suits choose 1 * (13 cards choose 5) = 5,148
Straight: There are 10 ways to choose a rank for the straight, and each rank can have 4 possible suits. Therefore, the number of straight combinations is:
10 ranks choose 1 * (4 suits choose 1)^5 - 36 = 10,200
Straight Flush: There are 10 ways to choose a rank for the straight flush, and each rank has 4 possible suits. The number of straight flush combinations is:
10 ranks choose 1 * 4 suits choose 1 = 40
Royal Flush: There are only 4 possible royal flush combinations, as there are only 4 different suits.
To calculate the probability, we divide the number of combinations for flushes or better by the total number of possible poker hands:
Probability = (5,148 + 10,200 + 40 + 4) / (52 cards choose 5) = 0.198%
Therefore, the chances of being dealt a flush or better are approximately 0.198%.
(c) To determine the chances of being dealt a hand that will lose to a straight, we need to subtract the probabilities of getting a straight or better from the total probability of getting any poker hand.
We've already calculated the probability of getting a straight in part (b) as 10,200 / (52 cards choose 5) = 0.0039%.
Total probability of getting any poker hand:
Total Probability = 1 - Probability of a straight or better
Therefore, Total Probability = 1 - 0.0039% = 99.9961%
Thus, the chances of being dealt a hand that will lose to a straight are approximately 99.9961%.