Question

Find the probability that when a hand of 8 cards is drawn from a well shuffled deck of 52 cards, it contains all Kings.

2

Answer 1

To find the probability that a hand of 8 cards drawn from a well-shuffled deck of 52 cards contains all 4 Kings, we can follow these steps:

1. **Identify the total number of ways to draw 8 cards from the deck:**
The total number of ways to choose 8 cards from a deck of 52 cards is given by the combination formula:

\[
\text{Total ways} = \binom{52}{8}
\]

2. **Determine the number of favorable outcomes:**
If all 4 Kings must be in the hand, we need to select 4 more cards from the remaining 48 cards (since we already have the 4 Kings). The number of ways to choose 4 cards from the remaining 48 cards is:

\[
\text{Favorable ways} = \binom{48}{4}
\]

3. **Calculate the probability:**
The probability \( P \) that a drawn hand of 8 cards contains all 4 Kings can be calculated using the formula:

\[
P(\text{all Kings}) = \frac{\text{Favorable ways}}{\text{Total ways}} = \frac{\binom{48}{4}}{\binom{52}{8}}
\]

4. **Calculate \(\binom{52}{8}\) and \(\binom{48}{4}\):**

- First, calculate \(\binom{52}{8}\):

\[
\binom{52}{8} = \frac{52!}{8! \times (52-8)!} = \frac{52 \times 51 \times 50 \times 49 \times 48 \times 47 \times 46 \times 45}{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} = 75287520
\]

- Now, calculate \(\binom{48}{4}\):

\[
\binom{48}{4} = \frac{48!}{4! \times (48-4)!} = \frac{48 \times 47 \times 46 \times 45}{4 \times 3 \times 2 \times 1} = 194580
\]

5. **Substituting the values into the probability formula:**

\[
P(\text{all Kings}) = \frac{194580}{75287520}
\]

To simplify this:

\[
P(\text{all Kings}) \approx 0.002585
\]

Thus, the probability that a hand of 8 cards contains all 4 Kings is approximately \( 0.002585 \) or \( 0.2585\% \).