Question

Four equally qualified runners, John, Bill, Ed, and Dave, run a 100 -meter sprint, and the order of finish is recorded. a. If the runners are equally qualified, what is the probability that Dave wins the race? b. What is the probability that Dave wins and John places second? c. What is the probability that Ed finishes last?

Short answer

Answer: The probability that Dave wins is 1/4 (25%); the probability that Dave wins and John places second is 1/12 (approx. 8.33%); and the probability that Ed finishes last is 1/4 (25%).

Step-by-step solution

Calculate the total number of possible outcomes

Since there are 4 runners and they all can finish in different positions, we calculate the total number of possible outcomes using the factorial: 4! = 4 × 3 × 2 × 1 = 24. So there are 24 different ways the runners can finish the race.

Calculate the probability that Dave wins (finishes first)

If Dave wins the race, there are 3 remaining runners that could finish in any order. We calculate the number of ways the other 3 runners can finish: 3! = 3 × 2 × 1 = 6. The probability that Dave wins is the number of ways Dave can win divided by the total number of possible outcomes: P(Dave wins) = \frac{6}{24} = \frac{1}{4}. So the probability that Dave wins the race is 1/4 or 25%.

Calculate the probability that Dave wins and John places second

If Dave wins and John places second, there are only two remaining positions for Bill and Ed. They can finish in any order: 2! = 2 × 1 = 2. The probability that Dave wins and John places second is the number of ways this scenario can happen divided by the total number of possible outcomes: P(Dave wins, John places second) = \frac{2}{24} = \frac{1}{12}. So the probability that Dave wins and John places second is 1/12 or approximately 8.33%.

Calculate the probability that Ed finishes last

If Ed finishes last, there are 3 remaining runners who can finish in any order: 3! = 3 × 2 × 1 = 6. The probability that Ed finishes last is the number of ways this scenario can happen divided by the total number of possible outcomes: P(Ed finishes last) = \frac{6}{24} = \frac{1}{4}. So the probability that Ed finishes last is 1/4 or 25%. In conclusion, the probability that Dave wins is 1/4; the probability that Dave wins and John places second is 1/12, and the probability that Ed finishes last is 1/4.

Key concepts

Factorial Calculations in Probability

Understanding factorial calculations is crucial when solving probability problems, particularly when dealing with permutations. A factorial, denoted by an exclamation point (!), is the product of all positive integers up to a given number. For instance, the factorial of 4, written as 4!, is calculated as 4 × 3 × 2 × 1, which equals 24.

In probability, factorial calculations help determine the total number of possible outcomes. For example, consider the exercise where four equally qualified runners—John, Bill, Ed, and Dave—are in a race. To determine the number of different ways the race can be concluded (the order of finish), we calculate the factorial of the number of runners, which in this case is 4! or 24. This is because the first place can be taken by any of the four, the second place by any of the remaining three, and so on. Whenever you encounter similar scenarios where order matters and there are no repetitions, factorial calculations will likely come into play.

Conditional Probability in Statistics

Conditional probability is a measure of the probability of an event occurring given that another event has already occurred. This concept is significant when dealing with dependent events in probability and statistics.

In the exercise mentioned, the solution to part (b) requires understanding conditional probability. After Dave wins, the race (the condition), what is the probability that John comes in second? Here, we already know Dave's outcome, and we're calculating the likelihood of John's placement based on that. Out of the 24 initial possible outcomes, knowing Dave finishes first reduces the possible outcomes for the remaining runners. We can then determine the probability that within the remaining possibilities, John takes the second place. Here, conditional probability allows for a refined calculation, considering a subset of the total possible outcomes.

Calculating Order of Finish Probability

The probability of a specific order of finish is the likelihood that a particular sequence of events will occur. When the sequence is crucial, such as determining the first, second, third, and last place in a race, we're dealing specifically with permutations of outcomes.

In our exercise, we explore the probability of various orders of finish. For instance, if we want to determine the probability that Ed finishes last, it's analogous to fixing Ed in the last place and looking at the permutations of the first three places among the other runners.

This is represented by the remaining three factorial, 3!, as there are six ways for the first three runners to finish, irrespective of the last place held by Ed. Hence, understanding how to calculate these probabilities is crucial for examining all possible scenarios and making informed decisions about the likelihood of these events.