Question

Player \(A\) has entered a golf tournament but it is not certain whether player \(B\) will enter. Player \(A\) has probability \(1 / 6\) of winning the tournament if player \(B\) enters and probability \(3 / 4\) of winning if player \(B\) does not enter the tournament. If the probability that player \(B\) enters is \(1 / 3,\) find the probability that player \(A\) wins the tournament.

Short answer

Answer: The probability that player A wins the tournament is 5/9.

Step-by-step solution

Find P(B')#

First, we need to find the probability that player \(B\) does not enter the tournament, \(P(B')\). Since there are only two possibilities (player \(B\) enters or does not enter), we can find \(P(B')\) using the complementary probability formula: $$P(B') = 1 - P(B)$$ Substitute the given value of \(P(B) = \frac{1}{3}\): $$P(B') = 1 - \frac{1}{3} = \frac{2}{3}$$

Apply the Law of Total Probability#

Now, we can use the law of total probability to find the probability that player \(A\) wins the tournament: $$P(W_A) = P(W_A|B)P(B) + P(W_A|B')P(B')$$ Substitute the given and calculated values: \(P(W_A|B) = \frac{1}{6}\), \(P(W_A|B') = \frac{3}{4}\), \(P(B) = \frac{1}{3}\), and \(P(B') = \frac{2}{3}\): $$P(W_A) = \frac{1}{6} \cdot \frac{1}{3} + \frac{3}{4} \cdot \frac{2}{3}$$

Calculate the Final Probability#

Finally, we can calculate the probability that player \(A\) wins the tournament: $$P(W_A) = \frac{1}{18} + \frac{1}{2} = \frac{1+9}{18} = \frac{10}{18}$$ Simplify the fraction to get the answer: $$P(W_A) = \frac{5}{9}$$ Thus, the probability that player \(A\) wins the tournament is \(\frac{5}{9}\).

Key concepts

Law of Total Probability

The Law of Total Probability is a fundamental concept in statistics and probability theory. It helps us determine the probability of an event by considering all possible scenarios. In our golf tournament exercise, it involves calculating the probability that player \(A\) wins, given the different entry possibilities for player \(B\).

The formula for the law of total probability is:

• \(P(W_A) = P(W_A|B) \cdot P(B) + P(W_A|B') \cdot P(B')\)

This equation shows:

• \(P(W_A|B)\) is the probability that player \(A\) wins if player \(B\) enters,
• \(P(B)\) is the probability that player \(B\) enters,
• \(P(W_A|B')\) is the probability that player \(A\) wins if player \(B\) does not enter, and
• \(P(B')\) is the probability that player \(B\) does not enter.

By substituting these values appropriately, we can find the overall probability that player \(A\) wins the tournament, taking into account both scenarios of player \(B\)'s participation. This powerful tool allows us to evaluate complex probabilities where multiple outcomes are possible.

Complementary Probability

Complementary probability is a simple yet powerful concept that helps to quickly find the probability of an event not occurring. In the context of the golf tournament, we are initially given the probability \(P(B)\) that player \(B\) enters the tournament. To solve the problem, we also need \(P(B')\), the probability that player \(B\) does not enter.

This is where complementary probability comes to our rescue. The idea is based on the fact that the sum of the probabilities of all possible outcomes must equal 1. Thus, the probability that something does not happen is simply:

• \(P(B') = 1 - P(B)\)

For our case:

• If \(P(B) = \frac{1}{3}\), then \(P(B') = 1 - \frac{1}{3} = \frac{2}{3}\).

This means there's a two-thirds probability that player \(B\) will not enter the tournament. Using the complementary probability simplifies our calculations and allows us to substitute this value into the law of total probability calculation.

Probability Calculation Steps

Now let's put it all together with the probability calculation steps used to solve our problem. Probability calculations can seem tricky at first, but breaking them down step by step always makes them manageable. Here's how we found the probability that player \(A\) wins the tournament:

• **Step 1**: Calculate \(P(B')\) using complementary probability: \(P(B') = 1 - \frac{1}{3} = \frac{2}{3}\).
• **Step 2**: Use the law of total probability to find \(P(W_A)\).
• **Step 3**: Substitute the values:
• \(P(W_A) = \frac{1}{6} \cdot \frac{1}{3} + \frac{3}{4} \cdot \frac{2}{3} = \frac{1}{18} + \frac{1}{2}\).
• **Step 4**: Simplify the expression:
• \(\frac{1}{18} + \frac{9}{18} = \frac{10}{18}\).
• Reduce \(\frac{10}{18}\) to \(\frac{5}{9}\).

This sequence of steps verifies that the probability that player \(A\) wins is \(\frac{5}{9}\). By following these organized steps, we ensure accuracy in our probability calculation.