[Math] Problem on Baye’s formula

probabilityprobability theory

I was reading A First Course in Probability by Sheldon Ross. I think I quite understood the below problem but I still feel fuzzy.

Problem: In answering on a multiple choice test, a student either know the answer or guesses. Let p be the probability that the students knows the answer and 1-p be the probability that the student guesses. Assume that a student who guesses at the answer will be correct with probability 1/m, where m is the number of multiple choice alternatives. What is the conditional probability that a student knew the answer to a question when he or she answered it correctly?

Solution as given in the book:
Let C be the event that the student answered the question correctly. And
Let K be the event that the student actually knows the answer.
$$P(K|C) = \frac{P(KC)}{P(C)}$$
$$ = \frac{P(C|K)P(K)}{P(C|K)P(K)+P(C|K^C)P(K^C)}$$
$$ = \frac{p}{p+(1/m)(1-p)}$$

Now this seems reasonable, only confusing is that how is

  1. Probability that the student knows the answer $= P(KC) = p$ but not $P(C|K)$
  2. While on the other hand the probability that a student who guesses the answer will be correct $= P(C|K^C) = 1/m$ but this time not $P(CK^C)$ or $P(K^CC)$.

I think its just that I am finding it difficult to determine the probabilities relation from the sentence formations.

  1. Is there any other simpler, non fuzzy approach to such problems?

Best Answer

Imagine that the test consists of $N$ questions, each with the same parameter $p$ of the student knowing the right answer; and assume that knowing the right answer on any question is independent of knowing the right answer on any other question.

In this scenario, each question will fall into one of three categories:

(A) questions where the student knew the answer (and hence and answered correctly);

(B) questions where the student didn't know but answered correctly; and

(C) questions where the student didn't know and answered incorrectly.

We would expect $pN$ questions to fall into category (A); and $(1-p)\cdot N\cdot \frac1m$ to fall into category (B).

Thus we expect $pN$ of the correctly answered questions to have been known. And so the probability of a correctly answered question to have been known is $$\frac{\#\text{known}}{\#\text{answered correctly}}=\frac{\#(A)}{\#(A)+\#(B)}$$

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