- The scenario is a geometric setting because it meets all four required conditions.
- Binary outcomes exist: landing on a "winner" segment is a success and landing on any other is a failure.
- Trials are independent: each spin of the wheel is independent of the previous ones.
- Trials are counted until the first success: the random variable \(X\) measures the spins until the "winner" is achieved.
- Probability is constant: the probability of success remains \(\frac{1}{4}\) for every trial.
- Final determination: \(\boxed{\text{Yes}}\)
Explanation
- Analysis of the Geometric Setting Criteria
To determine if a scenario is a geometric setting, we must verify if it satisfies four specific criteria related to a sequence of Bernoulli trials. A geometric setting arises when we perform independent trials until the first success occurs, with each trial having the same probability of success. In this case, Marshall spins a prize wheel with 4 equal segments, where landing on the "winner" segment constitutes a success. Let \(X\) be the number of spins until he lands on the "winner" segment.
- Evaluating the Four Geometric Conditions
We evaluate each condition for our scenario:
- Binary Outcomes: Each spin results in either a "winner" (Success) or "not a winner" (Failure). Since there are only these two possible categories, the outcomes are binary.
- Independent Trials: The outcome of any single spin of the prize wheel does not influence the outcome of any subsequent spin. Each spin is physically independent of the others.
- Counting Trials until Success: The random variable \(X\) is explicitly defined as the number of trials needed to achieve the first success. This is the definition of a geometric random variable.
- Constant Probability of Success: The wheel has 4 segments of equal size, and only one is labeled "winner." Therefore, the probability of success is always \(p = \frac{1}{4}\) for every spin. Since this value does not change, the probability of success is constant.
- Conclusion on the Geometric Setting
Because the scenario satisfies all four conditions—binary outcomes, independence, counting trials until success, and a constant probability of success—we conclude that this is indeed a geometric setting. Each spin is a Bernoulli trial, and we are counting the trials until the first success.
Examples
Geometric distributions are widely used in quality control and reliability engineering. For example, if a machine produces a defect with probability \(p\), the number of items produced until the first defective item appears follows a geometric distribution. This helps engineers estimate how many products can be manufactured before a maintenance check is required. If the probability of a defect is \(p = 0.05\), the expected number of items produced before the first defect is \(E[X] = \frac{1}{p} = \frac{1}{0.05} = 20\).