Questions: 5. Compute the probability that a 5 -card poker hand is dealt with 4 Aces.

Solution Steps

We need to compute the probability of being dealt a 5-card poker hand that contains exactly 4 Aces from a standard deck of 52 cards.

The hypergeometric distribution is defined as follows:

\[ P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}} \]

Where:

• \(N = 52\) (total number of cards),
• \(K = 4\) (total number of Aces),
• \(n = 5\) (number of cards drawn),
• \(k = 4\) (number of Aces we want in the hand).

We calculate the combinations needed for the hypergeometric formula:

1. \(\binom{K}{k} = \binom{4}{4} = 1\)
2. \(\binom{N-K}{n-k} = \binom{48}{1} = 48\)
3. \(\binom{N}{n} = \binom{52}{5} = 2598960\)

Substituting the values into the hypergeometric probability formula gives:

\[ P(X = 4) = \frac{\binom{4}{4} \cdot \binom{48}{1}}{\binom{52}{5}} = \frac{1 \cdot 48}{2598960} \]

Calculating the above expression results in:

\[ P(X = 4) = \frac{48}{2598960} \approx 0.0000185 \]

Final Answer