A student is getting ready to take an important oral...

A student is getting ready to take an important oral examination and is concerned about the possibility of having an "on" day or an "off" day. He figures that if he has an on day, then each of his examiners will pass him independently of each other, with probability $.8$, whereas, if he has an off day, this probability will be reduced to $.4$. Suppose that the student will pass the examination if a majority of the examiners pass him. If the student feels that he is twice as likely to have an off day as he is to have an on day, should he request an examination with 3 examiners or with 5 examiners?

Key Concepts

Conditional Probability

This concept involves determining the probability of an event occurring given that another event has occurred. In situations where outcomes depend on underlying conditions (such as having an 'on' or 'off' day), conditional probability allows us to update our calculations based on this known information to accurately assess overall likelihoods.

Binomial Distribution

The binomial distribution is used to model the number of successes in a fixed number of independent trials, each with the same probability of success. In exam settings where each examiner independently makes a decision (pass or fail), the total count of passes follows a binomial distribution which is critical for computing the probabilities associated with majority decisions.

Independence of Events

Independence means that the outcome of one event does not affect the outcome of another. This assumption is crucial when calculating combined probabilities across multiple examiners since it allows the simple multiplication of individual probabilities to determine the overall likelihood of a specific number of successes.

Majority Rule Decision

A majority rule decision criterion is used when an overall outcome is determined by the more than half the number of votes or approvals. In testing or committee decisions, the success criteria based on the majority of examiner votes necessitate comparing different scenarios to establish which setup maximizes the probability of achieving a favorable majority.

Decision Analysis Under Uncertainty

Decision analysis under uncertainty involves weighing different options when outcomes are not guaranteed and depend on probabilistic events. By considering different parameters, such as varying probabilities of success under different conditions and the structure of decision-making (for instance, the number of examiners), one can determine which option optimizes the chance of a positive outcome even in the face of uncertainty.

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