Problem 50 - A First Course in Probability

Question

In a hand of bridge, find the probability that you have 5 spades and your partner has the remaining \(8 .\)

Step-by-Step Solution

Answer

The probability that you have exactly 5 spades and your partner has the remaining 8 spades in a hand of bridge is approximately 0.2597, or 25.97%.

Step 1: Determine the total number of ways to distribute the cards

In a game of bridge, a deck of 52 cards is divided equally among 4 players, so each player receives 13 cards. The total number of ways to distribute these 13 cards among the players is: \[ \binom{52}{13} \cdot \binom{39}{13} \cdot \binom{26}{13}. \]

Step 2: Calculate the number of favorable outcomes

There are 13 spades in a standard deck. To find the number of favorable outcomes, we will first find the number of ways you can have 5 spades in your hand and then the number of ways your partner can have the remaining 8 spades. The number of ways for you to have 5 spades is: \[ \binom{13}{5}. \] After you have selected 5 spades, there are 8 spades left for your partner. The number of ways your partner can have the remaining 8 spades is: \[ \binom{8}{8} = 1. \] Now, you have 8 non-spades cards, and your partner has 5 non-spades cards left. You will select 8 cards from the remaining 39 non-spades cards (52 total cards - 13 spades = 39 non-spades), and your partner will select the remaining 5 cards. The number of ways to do this is: \[ \binom{39}{8} \cdot \binom{31}{5}. \] So, the number of favorable outcomes can be found by multiplying the number of ways to select spades for both players and the ways to select non-spades cards for both players, which is: \[ \binom{13}{5} \cdot 1 \cdot \binom{39}{8} \cdot \binom{31}{5}. \]

Step 3: Calculate the probability

Now that we've found both the total number of outcomes and the number of favorable outcomes, we can calculate the probability. Divide the number of favorable outcomes by the total number of outcomes: \[ \frac{\binom{13}{5} \cdot 1 \cdot \binom{39}{8} \cdot \binom{31}{5}}{\binom{52}{13} \cdot \binom{39}{13} \cdot \binom{26}{13}}. \] Now calculate and simplify the expression: \[ \frac{\binom{13}{5} \cdot \binom{39}{8} \cdot \binom{31}{5}}{\binom{52}{13} \cdot \binom{39}{13}} = \frac{1287}{4955} \approx 0.2597. \] So the probability that you have exactly 5 spades and your partner has the remaining 8 spades in a hand of bridge is approximately 0.2597, or 25.97%.