A poker hand consisting of 8 cards is dealt from a standard deck of 52 cards.

Find the probability that the hand contains exactly 4 face cards. Leave your answer as a reduced fraction.

The probability is 


To solve this problem, we need to calculate the probability of being dealt exactly 4 face cards in an 8-card poker hand from a standard deck of 52 cards. 

### Step 1: Determine the total number of possible hands

The number of ways to choose 8 cards from a deck of 52 cards is given by the combination formula:

\[
\binom{52}{8} = \frac{52!}{8!(52-8)!} = \binom{52}{8}
\]

### Step 2: Determine the number of successful hands (hands with exactly 4 face cards)

There are 12 face cards in a deck (4 Jacks, 4 Queens, and 4 Kings). We want to choose exactly 4 of these 12 face cards. The number of ways to choose 4 face cards from 12 is:

\[
\binom{12}{4} = \frac{12!}{4!(12-4)!} = \binom{12}{4}
\]

The remaining 4 cards must come from the 40 non-face cards in the deck. The number of ways to choose 4 non-face cards from the 40 non-face cards is:

\[
\binom{40}{4} = \frac{40!}{4!(40-4)!} = \binom{40}{4}
\]

### Step 3: Calculate the probability

The probability of getting exactly 4 face cards is the ratio of the number of successful hands to the total number of possible hands. This can be expressed as:

\[
\text{Probability} = \frac{\binom{12}{4} \cdot \binom{40}{4}}{\binom{52}{8}}
\]

Now, let's calculate the combinations and simplify.

#### Calculations:

\[
\binom{12}{4} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495
\]
\[
\binom{40}{4} = \frac{40 \times 39 \times 38 \times 37}{4 \times 3 \times 2 \times 1} = 91,390
\]
\[
\binom{52}{8} = \frac{52 \times 51 \times 50 \times 49 \times 48 \times 47 \times 46 \times 45}{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} = 752,875,680
\]

Now, calculate the probability:

\[
\text{Probability} = \frac{495 \times 91,390}{752,875,680} = \frac{45,276,050}{752,875,680}
\]

Simplifying the fraction:

\[
\text{Probability} = \frac{226,38025}{376,437,840} = \frac{22635}{18821892}
\







A poker hand consisting of 8 cards is dealt from a standard deck of 52 cards. Find the probability that the hand contains exactly 4 face cards. Leave your answer as a reduced fraction.

Explore how to calculate the probability of getting exactly 4 face cards in an 8-card poker hand from a standard deck of 52 cards. Learn step-by-step with combinatorics, including calculating card combinations and using the probability formula. Suitable for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation