A Flush
A flush in the card game of poker occurs if a player gets five cards that are all the same suit (clubs, diamonds, hearts, or spades). Answer the following questions to obtain the probability of being dealt a flush in five cards.


b. A flush can occur if we get five clubs or five diamonds or five hearts or five spades. Compute P(five clubs or five diamonds or five hearts or five spades). Note that the events are mutually exclusive.

Verified step by step guidance

1. Identify the total number of possible 5-card hands from a standard 52-card deck. This is given by the combination formula \(\binom{52}{5}\), which counts all possible 5-card combinations.
2. Calculate the number of ways to get five cards all from the same suit. Since each suit has 13 cards, the number of 5-card combinations from one suit is \(\binom{13}{5}\).
3. Since there are 4 suits (clubs, diamonds, hearts, spades), and the events of getting five cards from each suit are mutually exclusive, multiply the number of 5-card combinations from one suit by 4 to get the total number of flush hands.
4. Express the probability of getting a flush as the ratio of the total number of flush hands to the total number of possible 5-card hands. This is \(P(\text{flush}) = \frac{4 \times \binom{13}{5}}{\binom{52}{5}}\).
5. Interpret the result as the probability of being dealt a flush (five cards all of the same suit) in a 5-card hand from a standard deck.

Key Concepts

Probability of a Single Event

This concept involves calculating the likelihood of one specific outcome occurring, such as being dealt five cards all of the same suit (e.g., five clubs). It is found by dividing the number of favorable outcomes by the total number of possible outcomes in the sample space.

Mutually Exclusive Events

Mutually exclusive events cannot happen at the same time. In this problem, getting five clubs and five diamonds are mutually exclusive because a single hand cannot be both. This allows us to add their probabilities directly to find the total probability of any flush.

Addition Rule of Probability

The addition rule states that for mutually exclusive events, the probability of any one occurring is the sum of their individual probabilities. Here, the probability of a flush in any suit is the sum of the probabilities of flushes in clubs, diamonds, hearts, or spades.