Question
What is the probability of getting a royal flush in a game of poker?
Answers
GPT-4o mini
In a standard game of poker, a royal flush is the highest possible hand, consisting of the Ace, King, Queen, Jack, and Ten of the same suit.
To calculate the probability of being dealt a royal flush in a 5-card hand from a standard 52-card deck, we can use the following method:
1. **Number of ways to get a royal flush**: There are 4 different suits (hearts, diamonds, clubs, and spades), and each suit can form exactly 1 royal flush. Therefore, there are a total of 4 royal flushes.
2. **Total number of 5-card combinations**: The number of ways to choose 5 cards from a 52-card deck can be calculated using combinations. This is given by the formula:
\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]
where \( n \) is the total number of items (52 cards) and \( k \) is the number of items to choose (5 cards):
\[
\binom{52}{5} = \frac{52!}{5!(52-5)!} = \frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1} = 2,598,960
\]
3. **Calculating the probability**: The probability \( P \) of being dealt a royal flush can be calculated as the number of favorable outcomes (4) divided by the total number of possible 5-card hands (2,598,960):
\[
P(\text{royal flush}) = \frac{4}{2,598,960} \approx 0.0000015391
\]
In percentage terms, the probability of being dealt a royal flush is:
\[
P(\text{royal flush}) \approx 0.000154\%
\]
Thus, the overall probability of being dealt a royal flush in a standard game of poker is about 1 in 649,740.
To calculate the probability of being dealt a royal flush in a 5-card hand from a standard 52-card deck, we can use the following method:
1. **Number of ways to get a royal flush**: There are 4 different suits (hearts, diamonds, clubs, and spades), and each suit can form exactly 1 royal flush. Therefore, there are a total of 4 royal flushes.
2. **Total number of 5-card combinations**: The number of ways to choose 5 cards from a 52-card deck can be calculated using combinations. This is given by the formula:
\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]
where \( n \) is the total number of items (52 cards) and \( k \) is the number of items to choose (5 cards):
\[
\binom{52}{5} = \frac{52!}{5!(52-5)!} = \frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1} = 2,598,960
\]
3. **Calculating the probability**: The probability \( P \) of being dealt a royal flush can be calculated as the number of favorable outcomes (4) divided by the total number of possible 5-card hands (2,598,960):
\[
P(\text{royal flush}) = \frac{4}{2,598,960} \approx 0.0000015391
\]
In percentage terms, the probability of being dealt a royal flush is:
\[
P(\text{royal flush}) \approx 0.000154\%
\]
Thus, the overall probability of being dealt a royal flush in a standard game of poker is about 1 in 649,740.