When answering a question on a multiple choice test, a...

When answering a question on a multiple choice test, a student is given 5 choices, one of which is correct. The test is so designed that the choices are very close and the probability of getting the correct answer, when you know the material, is $0.6 .$ In a class where $70 \%$ of the students are well prepared, a randomly chosen student answers the question correctly. What is the probability that the student really knew the material?

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Key Concepts

Conditional Probability

This concept involves finding the likelihood of an event occurring given that another event has occurred. It forms the foundation for analyzing problems where outcomes depend on related events, such as determining the probability that a student was well prepared given a correct answer.

Bayes' Theorem

Bayes' Theorem is used to update the probability of a hypothesis based on new evidence. In this context, it helps calculate the probability that a student knew the material (the hypothesis) given that they answered correctly (the evidence), by considering the initial likelihood of being well prepared and the probability of answering correctly under each circumstance.

Total Probability Theorem

This theorem is used to compute the overall probability of an event by summing over the probabilities of several mutually exclusive scenarios that lead to the event. It assists in breaking down the overall probability of a correct answer into the contributions from both well-prepared and unprepared students.

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