Questions: Five cards are drawn randomly from a standard deck of 52 cards. Determine the probability that exactly 3 of these cards are Aces. Write your answer in decimal form, rounded to 5 decimal places.

Solution Steps

To determine the probability of drawing exactly 3 Aces from a standard deck of 52 cards, we use the hypergeometric distribution formula:

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

Substituting the values \(N = 52\), \(K = 4\), \(n = 5\), and \(k = 3\):

\[ P(X = 3) = \frac{\binom{4}{3} \binom{48}{2}}{\binom{52}{5}} = \frac{4 \cdot 1128}{2598960} \approx 0.00174 \]

The mean \(\mu\) of the hypergeometric distribution is given by:

\[ \mu = n \cdot \frac{K}{N} \]

Substituting the values:

\[ \mu = 5 \cdot \frac{4}{52} = 0.38462 \]

The variance \(\sigma^2\) is calculated using the formula:

\[ \sigma^2 = n \cdot \frac{K}{N} \cdot \frac{N-K}{N} \cdot \frac{N-n}{N-1} \]

Substituting the values:

\[ \sigma^2 = 5 \cdot \frac{4}{52} \cdot \frac{48}{52} \cdot \frac{47}{51} \approx 0.32718 \]

The standard deviation \(\sigma\) is the square root of the variance:

\[ \sigma = \sqrt{0.32718} \approx 0.572 \]

Final Answer