Questions: A five cards hand is dealt at random from a standard deck. What is the probability that the hand contains exactly two red cards? Round your answer to the nearest hundredth.

Solution Steps

We need to find the probability of drawing exactly two red cards from a five-card hand dealt from a standard deck of 52 cards, which contains 26 red cards and 26 black cards.

The probability of drawing exactly \( k \) successes (red cards) in \( n \) draws (total cards drawn) from a population of \( N \) items (total cards) containing \( K \) successes (total red cards) is given by the hypergeometric distribution:

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

For our problem:

• \( N = 52 \) (total cards)
• \( K = 26 \) (total red cards)
• \( n = 5 \) (cards drawn)
• \( k = 2 \) (red cards drawn)

Substituting these values into the formula gives:

\[ P(X = 2) = \frac{\binom{26}{2} \binom{26}{3}}{\binom{52}{5}} \]

Now we calculate the combinations:

• \( \binom{26}{2} = \frac{26 \times 25}{2 \times 1} = 325 \)
• \( \binom{26}{3} = \frac{26 \times 25 \times 24}{3 \times 2 \times 1} = 2600 \)
• \( \binom{52}{5} = \frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1} = 2598960 \)

Now substituting the values of the combinations back into the probability formula:

\[ P(X = 2) = \frac{325 \times 2600}{2598960} \]

Calculating this gives:

\[ P(X = 2) = \frac{845000}{2598960} \approx 0.325 \]

Rounding the result to the nearest hundredth, we find:

\[ P(X = 2) \approx 0.33 \]

Final Answer