Problem 31 - A First Course in Probability

Question

A 3 -person basketball team consists of a guard, a forward, and a center. (a) If a person is chosen at random from each of three different such teams, what is the probability of selecting a complete team? (b) What is the probability that all 3 players selected play the same position?

Step-by-Step Solution

Answer

The probabilities for the given question are as follows: (a) The probability of selecting a complete team consisting of one guard, one forward, and one center is \(\frac{2}{9}\). (b) The probability of selecting all 3 players with the same position (all guards, all forwards, or all centers) is \(\frac{1}{9}\).

Step 1: Part (a) Step 1: Calculate the total number of possible outcomes

We need to find the total number of ways we can select one player from each of the three teams. As each team has three players, we have 3 choices from the first team, 3 choices from the second team, and 3 choices from the third team. The total number of possible outcomes is the product of these choices: \(3 \times 3 \times 3 = 27\)

Step 2: Part (a) Step 2: Calculate the number of successful outcomes

To select a complete team, we have to select one guard, one forward, and one center. This can be done in the following way: 1. Choose a guard from the first team, a forward from the second team, and a center from the third team. 2. Choose a guard from the first team, a center from the second team, and a forward from the third team. 3. Choose a forward from the first team, a guard from the second team, and a center from the third team. 4. Choose a forward from the first team, a center from the second team, and a guard from the third team. 5. Choose a center from the first team, a guard from the second team, and a forward from the third team. 6. Choose a center from the first team, a forward from the second team, and a guard from the third team. Thus, we have 6 successful outcomes.

Step 3: Part (a) Step 3: Calculate the probability of selecting a complete team

The probability of selecting a complete team is the number of successful outcomes divided by the total number of possible outcomes: \(\frac{6}{27} = \frac{2}{9}\)

Step 4: Part (b) Step 1: Calculate the number of successful outcomes for selecting the same position

To select all players with the same position: 1. Choose a guard from each team: there's only 1 way to do this. 2. Choose a forward from each team: there's only 1 way to do this. 3. Choose a center from each team: there's only 1 way to do this. Thus, there are a total of 3 successful outcomes for selecting the same position.

Step 5: Part (b) Step 2: Calculate the probability of all 3 players being of the same position

The probability of selecting all players with the same position is the number of successful outcomes divided by the total number of possible outcomes: \(\frac{3}{27} = \frac{1}{9}\) The probabilities for the two parts of the exercise are: (a) The probability of selecting a complete team is \(\frac{2}{9}\) (b) The probability of selecting all players with the same position is \(\frac{1}{9}\)