Question

Under the "no pass, no play" rule for athletes, an athlete who fails a course cannot participate 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 before 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 0.255 or 25.5%.

Step-by-step solution

Identify the given probabilities and percentage

We are given the following information: - Probability of disqualification for an athlete with no previous disqualification (P(DQ | Never DQ before)): 0.15 - Probability of disqualification for an athlete with a previous disqualification (P(DQ | DQ before)): 0.5 - Percentage of athletes who have been disqualified before: 30% or 0.3

Calculate the probability of never been disqualified before

We are given the percentage of athletes who have been disqualified before. To find the percentage of athletes who have never been disqualified before, we need to subtract the given percentage from 100%, or 1 - 0.3 = 0.7. So, the probability of never been disqualified before is 0.7.

Use the law of total probability

The law of total probability states that the unconditional probability can be found by multiplying the probability of each event by the probability of the event happening and summing them. Unconditional probability of disqualification (P(DQ)) can be calculated as follows: P(DQ) = P(DQ | Never DQ before) × P(Never DQ before) + P(DQ | DQ before) × P(DQ before)

Calculate the unconditional probability of disqualification

Using the formula from step 3 and the probabilities from step 1 and 2, we can find the unconditional probability of disqualification: P(DQ) = 0.15 × 0.7 + 0.5 × 0.3 P(DQ) = 0.105 + 0.15 P(DQ) = 0.255 The unconditional probability that an athlete will be disqualified during the next grading period is 0.255 or 25.5%.

Key concepts

Law of Total Probability

In probability theory, the Law of Total Probability is a fundamental rule that relates marginal probabilities to conditional probabilities. This law helps us find the overall probability of an event, known as the unconditional probability, by considering all possible scenarios that could lead to that event.

Imagine you have a bag of red and blue marbles, but also, within this bag, there are smaller bags dividing the marbles by their size — large and small. If you want to know the probability of randomly picking a red marble, regardless of its size, you would apply the Law of Total Probability. You calculate the chances of picking a red marble from each size category (conditional probabilities) and then combine these probabilities, weighted by the likelihood of selecting a marble from each size category (their marginal probabilities).

Here’s the equation for this law:
\[ P(A) = \sum P(A \mid B_i)P(B_i) \]
where:

• \(P(A)\) is the unconditional probability of event A,
• \(P(A \mid B_i)\) is the conditional probability of A given \(B_i\),
• \(P(B_i)\) is the probability of the sub-event \(B_i\),
• The \(B_i's\) are the disjoint events that cover the entire sample space.

In the example of the athletes, we considered two scenarios: those who have never been disqualified before and those who have. We multiplied the probability of disqualification for each group by the probability of being in that group, and added the results together to find the total probability of disqualification.

Probability Theory

At the core of decision-making and scientific analysis lies Probability Theory, a branch of mathematics that deals with the analysis of random events. The fundamental premise of probability theory is to quantify the uncertainty of events using a probability, a value between 0 and 1, where 0 means an event is impossible, and 1 means it is certain.

This theory provides the mathematical framework for understanding and working with uncertainty and variability in everything from games of chance to weather forecasting, finance, and medical research. In context with our athlete’s example:

• When we say there is a probability of 0.15 that an athlete who has never been disqualified will be disqualified, we are predicting the likelihood of a single, uncertain event.
• The unconditional probability of 0.255 for an athlete being disqualified doesn’t mean that 25.5 athletes out of 100 will be disqualified; it means that each athlete has a 25.5% chance of being disqualified.

In essence, probability theory allows us to make informed guesses about the future based on the distribution of past events. The theory emphasizes the collection, analysis, interpretation, and presentation of data which could lead to the statistical disqualification of certain assumptions when they do not align with the observed probabilities.

Statistical Disqualification

The term Statistical Disqualification is not commonly used in probability or statistics. However, it likely refers to the process of determining when a particular assumption or model is not supported by the data. This concept is essential when considering conditional and unconditional probabilities, as it helps in refining the models by disqualifying the invalid assumptions.

In statistical hypothesis testing, a hypothesis might be disqualified if the observed data is very unlikely under that hypothesis. For example, if we assumed that previous disqualification does not affect the chances of future disqualification, but the data showed a significant difference in disqualification rates, we would statistically disqualify that assumption.

In our scenario:

• We are implicitly assuming that past disqualification is a relevant factor for future disqualification chances.
• If the data repeatedly showed no difference between the disqualification rates for those previously disqualified and not, that notion would be statistically disqualified.

Therefore, statistical disqualification acts as a corrective mechanism that ensures our models and predictions remain valid and based on current evidence.