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%.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics dealing with counting, arrangement, and combination of objects. It is central to probability theory because it provides the tools to calculate possible outcomes. When you calculate probabilities in card games, like bridge, combinatorics allows you to count the exact number of ways you can arrange cards.

• The notion of combinations is important as it helps to calculate the number of ways to choose objects from a larger set without caring about their order.
• The formula for combinations is \[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]where \(n\) is the total number of items, \(r\) is the number of items to be chosen, and \(!\) denotes the factorial, which means the product of all positive integers up to that number.

To illustrate, if there are 13 spades and you want to choose 5, you apply the formula \(\binom{13}{5}\) to know the number of ways to do that.

Bridge Game

Bridge is a strategic card game played by four players. It uses a standard deck of 52 cards, divided into four suits: spades, hearts, diamonds, and clubs. Each player is dealt 13 cards, making up their hand.

• The key to bridge is in the distribution of suits across the players. To calculate the odds of getting certain cards, like exactly 5 spades in your hand, probability theory, particularly combinatorics, is used.
• Bridge is a partnership game, meaning you work with a partner to achieve your shared goal of winning the most tricks, which are rounds of play in which each player plays a card.

Understanding how cards can be distributed among players is crucial to developing the strategy necessary for effective gameplay.

Card Distribution

Card distribution refers to the possible ways cards can be dealt or divided among players. Each game of bridge starts with distributing cards, where each player receives an equal share of the deck—13 cards each in a four-player game.

• Calculating probability based on card distribution involves determining all the ways cards can be dealt and how likely it is for certain cards to end up in specific hands.
• In our example, you want 5 spades, and your partner wants the remaining 8 spades in a game of bridge. To find this probability, you calculate the combinations of ways to initially get the cards you and your partner want.

The process also involves considering the remaining cards to complete each hand, which requires an understanding of the entire deck's composition. Combining the use of combinatorics with the constraints of the game rules allows you to find the probability of specific distributions.