Question

Five cards are selected from a 52-card deck for a poker hand. a. How many simple events are in the sample space? b. A royal flush is a hand that contains the \(\mathrm{A}, \mathrm{K}, \mathrm{Q}, \mathrm{J},\) and 10 , all in the same suit. How many ways are there to get a royal flush? c. What is the probability of being dealt a royal flush?

Short answer

Answer: The probability of getting a royal flush in a 5-card poker hand is approximately 0.000154% or 1.54 × 10^-6.

Step-by-step solution

a. Finding the total number of simple events in the sample space

To find the total number of simple events (5-card poker hands) in the sample space, we need to determine the number of ways to choose 5 cards from a 52-card deck. This can be found using the combination formula: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where \(n\) is the number of items and \(k\) is the number of items to choose. In this case, \(n = 52\) and \(k = 5\). So, the total number of simple events is \(\binom{52}{5} = \frac{52!}{5!(52-5)!} = 2,598,960\).

b. Finding the number of ways to get a royal flush

A royal flush is a poker hand consisting of the ace, king, queen, jack, and ten, all of the same suit. There are four possible suits: clubs, diamonds, hearts, and spades. Since there are no other restrictions, there is only one way to choose the cards for each suit, making a total of 4 ways to get a royal flush.

c. Calculating the probability of a royal flush

To calculate the probability of a royal flush, we need to divide the number of ways to get a royal flush by the total number of simple events in the sample space. From our previous calculations, we know there are 4 ways to get a royal flush and 2,598,960 simple events. So, the probability of getting a royal flush is \(\frac{4}{2,598,960} \approx 1.54 \times 10^{-6} \approx 0.000154\%\).

Key concepts

Combinatorics

Combinatorics is the mathematics of counting and arrangement. It's essential for calculating possibilities in various scenarios. When dealing with card games like poker, we often need to determine the different combinations or arrangements possible with a set number of items. For example, to find out how many ways we can select 5 cards from a 52-card deck, we use combinations because the order in which cards are dealt doesn't matter.
The formula for combinations is:

• Combination formula: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

Here, \( n \) represents the total number of items (52 cards), and \( k \) represents the number of items to choose (5 cards).
Applying these values: \( \binom{52}{5} = \frac{52!}{5!(52-5)!} = 2,598,960 \). This result tells us there are 2,598,960 different ways to draw a hand of 5 cards from a deck of 52.

Sample Space

The sample space in probability refers to the set of all possible outcomes of a random experiment. In the context of a poker hand, the sample space includes every possible combination of 5 cards that can be drawn from a deck of 52 cards.
For a 5-card poker hand, the sample space consists of all the different hands you can draw from the deck, which we calculated as 2,598,960 possible combinations. Understanding sample space is crucial because it serves as the denominator in probability calculations, representing all the potential simple events that could occur in a given scenario.
Each unique arrangement of 5 cards counts as one simple event, whether it’s a royal flush, a straight, or any other hand. This large number highlights the vast number of ways cards can be combined, giving poker its complexity and intrigue.

Poker Hand Probability

Poker hand probability involves calculating the likelihood of drawing specific hands from a deck of cards. In this case, we're interested in finding the probability of being dealt a royal flush, one of the rarest hands in poker.
To find this probability, we take the number of favorable outcomes (ways to get a royal flush) and divide it by the total number of possible poker hands. As calculated, there are 4 ways to draw a royal flush—one for each suit.

• Probability calculation: \( \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{4}{2,598,960} \)

This calculation yields approximately \( 1.54 \times 10^{-6} \), or about 0.000154%. This extraordinarily low probability demonstrates just how unlikely it is to be dealt a royal flush, explaining its prestigious status in poker.