Question

Under the "no pass, no play" rule for athletes, an athlete who fails a course is disqualified from participating in sports activities during the next grading period. Suppose the probability that an athlete who has not previously been disqualified will be disqualified is .15 and the probability that an athlete who has been disqualified will be disqualified again in the next time period is \(.5 .\) If \(30 \%\) of the athletes have been disqualified before, what is the unconditional probability that an athlete will be disqualified during the next grading period?

Short answer

Answer: The unconditional probability that an athlete will be disqualified during the next grading period is 25.5%.

Step-by-step solution

Identify the variables and probabilities

Let's define the following events: - A: An athlete is disqualified during the next grading period - B: An athlete has been disqualified before We are given the following probabilities: - P(A|~B) = 0.15 (probability of disqualification for athletes who have not been disqualified before) - P(A|B) = 0.5 (probability of disqualification for athletes who have been disqualified before) - P(B) = 0.3 (probability that an athlete has been disqualified before)

Calculate the probability of an athlete not being disqualified before

First, we need to find the probability of an athlete not being disqualified before (P(~B)). We know that 30% of the athletes have been disqualified before (P(B) = 0.3), so the probability of an athlete not being disqualified before is the complement of that: P(~B) = 1 - P(B) = 1 - 0.3 = 0.7

Use the Law of Total Probability

Now we can use the Law of Total Probability to find the unconditional probability of an athlete being disqualified during the next grading period (P(A)): P(A) = P(A|B) * P(B) + P(A|~B) * P(~B)

Plug in probabilities and find the result

Substitute the given probabilities and calculate the unconditional probability of disqualification: P(A) = (0.5 * 0.3) + (0.15 * 0.7) = 0.15 + 0.105 = 0.255 So, the unconditional probability that an athlete will be disqualified during the next grading period is 25.5%.

Key concepts

Conditional Probability

Conditional probability is a vital concept in probability theory. It describes the probability of an event occurring, given that another event has already happened. To better understand, think about flipping a coin two times. If you want to know the probability of getting heads on the second flip, given that the first flip was also heads, you are looking for a conditional probability.

In the context of our exercise, we used conditional probabilities when we considered athletes with different histories of disqualification. The probability data was given as follows:

• P(A|~B) = 0.15: This means there is a 15% chance of disqualification if the athlete has not been disqualified before.
• P(A|B) = 0.5: This indicates a 50% chance of disqualification if the athlete has previously been disqualified.

By managing these conditional probabilities, we could apply them to real-world scenarios, such as predicting future disqualification using past behavior.

Probability Theory

Probability theory is the mathematical study of random events. It helps us understand how likely events will happen. With probability theory, we can analyze and model complex systems where certainty is hard to find. It's often used in a variety of fields like finance, sports, and science.

At the heart of the exercise, we see probability theory at work. Here's how:

• Event A: Representing an athlete being disqualified in the future.
• Event B: Showing a past disqualification.

Each event has its probability, and they came together through the broader principle of probability theory. By knowing how these events fit into probability theory, you can make accurate predictions or decisions based on calculated likelihoods.

Statistical Analysis

Statistical analysis uses detailed methods and techniques to interpret data and identify relationships. In our exercise, we use statistical analysis to determine the overall probability of athlete disqualification by combining given conditional probabilities and event data.

Here's a breakdown of how statistical analysis is applied:

• Start by identifying known probabilities about athlete disqualifications.
• Calculate the likelihood of past disqualification using complementary probabilities: 30% disqualified, 70% not disqualified.
• Employ the Law of Total Probability to determine the overall likelihood.

In this exercise, the Law of Total Probability allowed us to methodically organize and compute the unconditional probability of an event considering two different groups of athletes. This form of analysis provides clarity in prediction and decision-making and is crucial in areas like sports management and policy-making.