Question

Compute the probability of being dealt at random and without replacement a 13 -card bridge hand consisting of: (a) 6 spades, 4 hearts, 2 diamonds, and 1 club; (b) 13 cards of the same suit.

Short answer

The probability for scenario (a) is \( \frac{\binom{13}{6} * \binom{13}{4} * \binom{13}{2} * \binom{13}{1}}{\binom{52}{13}} \) and for scenario (b) it is \( \frac{4 * \binom{13}{13}}{\binom{52}{13}} \).

Step-by-step solution

Total number of possible hands

First, determine how many total different 13 card hands can be dealt from a deck of 52 cards. This can be calculated using combinations as \( \binom{52}{13} \).

Compute probability for scenario (a)

For scenario (a), calculate the chances of getting 6 spades from 13, 4 hearts from 13, 2 diamonds from 13, and 1 club from 13. This is done by calculating the combinations for each and then multiplying them together. The calculation is as follows: \( \binom{13}{6} * \binom{13}{4} * \binom{13}{2} * \binom{13}{1} \). Divide this product by the total number of possible hands from Step 1 to get the probability.

Compute probability for scenario (b)

For scenario (b), calculate the chances of getting all 13 cards of the same suit. This means: \( \binom{13}{13} \) for each suit. Then, since this can occur for any of the 4 suits, multiply by 4. Divide this product by the total number of possible hands from Step 1 to get the probability.