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Chapter 8: Problem 48

A soccer player takes repeated shots on goal. What assumption must we make if we want to model a succession of shots by a player as a sequence of Bernoulli trials?

Short Answer

Expert verified
To model the soccer player's repeated shots on goal as a sequence of Bernoulli trials, we must make the assumption that each shot is an independent event, and the probability of success (scoring a goal) and failure (missing the goal) remain constant throughout all the trials. In other words, the outcome of one shot does not affect the outcome of any other shot, and the player's ability to score does not change during the series of shots.

Step by step solution

01

Identify the outcomes of each trial

In this situation, the two possible outcomes for each trial are scoring a goal (success) and missing the goal (failure). Ensure that only two outcomes are possible for each shot on goal.
02

Determine the probability of success

For each shot on goal, there is a fixed probability (p) that the player will score. This probability should remain constant throughout all the trials, meaning the player's ability to score should not change during the series of shots.
03

Determine the probability of failure

Similarly, the probability of missing the goal (failure) must be fixed as well. In Bernoulli trials, the probabilities of success (p) and failure (q) always add up to 1, so q = 1 - p, meaning the probability of missing should also remain constant throughout all trials.
04

Verify trial independence

The assumption we must make to model this situation as a series of Bernoulli trials is that each shot on goal is an independent event, meaning the outcome of one shot does not affect the outcome of any other shot. In other words, scoring a goal in one trial does not make the player more or less likely to score in the next trial. To sum up, the assumption we need to make in this soccer exercise is that the shots on goal are independent events, and the probability of success (scoring) and failure (missing) remain constant throughout all the trials.

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