Final answer:
A combination of combinatorial formulas and the understanding of poker rules are applied. Each hand's probability is derived by counting the specific arrangements that constitute the hand and dividing by the total number of 5-card combinations possible from a deck of 52 cards.
Step-by-step explanation:
To calculate the probability of different poker hands, we need to consider the number of favorable outcomes (the number of ways we can get that hand) and the total number of possible outcomes (the total number of ways we can draw five cards from a standard deck of 52 cards).
Let's go through each case:
(a) Straight Flush:
Formula:
P(Straight Flush)= Number of Straight Flush Hands/Total Possible Hands
Details:
Number of Straight Flush Hands: There are 10 possible straight flushes in a deck (one for each possible starting card in a straight).
Total Possible Hands:
(52 5), which is the number of ways to choose 5 cards from a deck of 52.
(b) Royal Flush:
Formula:
P(Royal Flush)= Number of Royal Flush Hands/ Total Possible Hands
Details:
Number of Royal Flush Hands: There are 4 possible royal flushes (one for each suit).
Total Possible Hands: Same as above.
(c) Three of a Kind:
Formula:
P(Three of a Kind)= Number of Three of a Kind Hands/ Total Possible Hands
Details:
Number of Three of a Kind Hands: Choose a rank (13 choices), then choose 3 suits from that rank (4 choices each).
Total Possible Hands: Same as above.
(d) Four of a Kind:
Formula:
P(Four of a Kind)= Number of Four of a Kind Hands/Total Possible Hands
Details:
Number of Four of a Kind Hands: Choose a rank (13 choices), then choose 1 suit from that rank (4 choices).
Total Possible Hands: Same as above.
(e) Straight:
Formula:
P(Straight)= Number of Straight Hands / Total Possible Hands
Details:
Number of Straight Hands: There are 10×4⁵ possible straights (10 possible starting cards, 4 suits for each card).
Total Possible Hands: Same as above.
(f) Bust:
Formula:
P(Bust)= Number of Bust Hands/ Total Possible Hands
Details:
Number of Bust Hands: Choose 5 cards that do not form any of the specified poker hands.
Total Possible Hands: Same as above.
These formulas are based on basic principles of combinatorics and probability. The specific calculations involve counting the number of ways to get each type of hand and dividing by the total number of possible hands.