Question

A multiple choice exam has 4 choices for each question. A student has studied enough so that the probability they will know the answer to a question is 0.5, the probability that they will be able to eliminate one choice is 0.25, otherwise all 4 choices seem equally plausible. If they know the answer they will get the question right. If not, they have to guess from the 3 or 4 choices. As the teacher, you want to test to measure what the student knows. If the student answers a question correctly, what is the probability they knew the answer?

Answer 1

To find the probability that the student knew the answer to a question given that they answered it correctly, we use Bayes' theorem. The probability is 0.8, or 80%.

Step-by-step explanation:

To find the probability that the student knew the answer to a question given that they answered it correctly, we need to use Bayes' theorem. Let's define the events:

A: Knowing the answer to a question

B: Answering a question correctly

We are given that the probability of knowing the answer is 0.5 (P(A) = 0.5) and the probability of answering a question correctly given that the student knows the answer is 1 (P(B|A) = 1). We are asked to find P(A|B), the probability of knowing the answer given that the student answered the question correctly.

Using Bayes' theorem, we have:

P(A|B) = P(B|A) * P(A) / P(B)

Substituting the given values, we get:

P(A|B) = 1 * 0.5 / P(B)

We need to find P(B), the probability of answering a question correctly. This probability depends on whether the student knew the answer or not. If the student knew the answer, the probability of answering correctly is 1. If the student didn't know the answer, the probability of randomly guessing correctly is 1/4 (since there are 4 choices).

Let's assume that the student knows the answer with probability 0.5 and didn't know the answer with probability 0.5. Then, the probability of answering correctly can be calculated as:

P(B) = P(B|A) * P(A) + P(B|A') * P(A')

Substituting the given values, we get:

P(B) = 1 * 0.5 + (1/4) * 0.5 = 0.625

Now we can substitute this value back into Bayes' theorem to find P(A|B):

P(A|B) = 1 * 0.5 / 0.625 = 0.8

So, the probability that the student knew the answer given that they answered the question correctly is 0.8, or 80%.

Answer 2

probability that the student knew the answer given that he answered the question correctly is 0.7742 (77.42%)

Step-by-step explanation:

a student can get the question right in 3 ways:

- knowing the answer with probability 0.5

- eliminating one of the 4 choices and guessing with the remaining 3 with probability 0.25

- or guessing from the 4 choices with probability 0.25

then defining the event R= getting the answer right , we have

P(R)= probability of knowing the answer*probability of getting the question right if knowing the answer + probability of eliminating one answer* probability of getting the question right if eliminates one answer + probability of guessing the 4 choices * probability of getting the question right if guessing the 4 choices

thus

P(R)= 0.5*1 + 0.25* 1/3 + 0.25*1/4 = 0.6458

then we use conditional probability through the theorem of Bayes. Defining K= student knew the answer

then

P(K/R) = P(K∩R) /P(R) = 0.5*1/0.6458 = 0.7742 (77.42%)

where

P(K∩R) = probability that the student knew the answer and answers the question correctly

P(K/R)= probability that the student knew the answer given that he answered the question correctly