Final answer:
The correct distribution for the scenario where many people can run a short distance but fewer can run a longer distance is the exponential distribution. It models events that occur continuously and independently at a constant rate, and the probability decreases exponentially as the distance increases.
Step-by-step explanation:
The Exponential Distribution
The question asks to identify which of the following distributions is apt for a scenario where many people can run a short distance of under two miles, but as the distance increases, fewer people can run that far. The correct distribution that describes this situation is the exponential distribution. This distribution is often used to model the time elapsed between events that happen continuously and independently at a constant average rate. In the context of running, as the distance increases, the time or the effort required also increases, and hence the probability of people being able to run that distance decreases exponentially.
To further provide insight into the options provided, the binomial distribution is used for scenarios with two possible outcomes (e.g., success or failure) for each trial, the uniform distribution is where all outcomes are equally likely, and the normal distribution involves symmetric data around a central mean.
In contrast, the exponential distribution is characterized by a rapid decrease in probability as the variable of interest increases. It is suitable for describing the running scenario because there's a high likelihood of many people running a short distance, but the likelihood decreases sharply as the distance gets longer. This type of distribution is complemented with the memoryless property where the probability of an event occurring in the next time period is independent of when the last event occurred.