Question

In an exam, there is a problem that 60% of students know the correct answer. However, there is 15% chance that a student picked the wrong answer even if he/she knows the right answer and there is also a 25% chance that a student does not know the right answer but guessed it correctly. If a student did get the problem right, what is the probability that this student really knows the answer?

Answer 1

P(students know the correct answer) = 0.6

P(picked the wrong answer | student knows the right answer) = 0.15

P(picked the right answer | student knows the right answer) = 1 - P(picked the wrong answer | student knows the right answer) = 1 - 0.15 = 0.85

P(picked the right answer | student doesn't know the right answer) = 0.25

P(get the problem right) = P(picked the right answer | student knows the right answer) * P(student knows the right answer) + P(picked the right answer | student doesn't know the right answer) * P(student doesn't know the right answer)

                                      = 0.85 * 0.6 + 0.25 * (1 - 0.6)

                                      = 0.61

P(student knows the right answer | get the problem right) = P(picked the right answer | student knows the right answer) * P(student knows the right answer) / P(get the problem right)

                                                                                           = 0.85 * 0.6 / 0.61

                                                                                           = 0.8361 (ans)