A student takes an examination consisting of 20 true-false questions. The student knows the answer to [math]N[/math] of the questions, which are answered correctly, and guesses the answers to the rest. The conditional probability that the student knows the answer to a question, given that the student answered it correctly, is 0.824. Calculate [math]N[/math]

• 8
• 10
• 14
• 16
• 18

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: C

Let C and K denote respectively the event that the student answers the question correctly and the event that he actually knows the answer. The known probabilities are

[[math]] \operatorname{P}(C | K^c) = 0.5, \, \operatorname{P}(C | K ) = 1, \, \operatorname{P}(K | C) = 0.824, \, \operatorname{P}(K) = N/20. [[/math]]

Then,

[[math]] \begin{align*} 0.824 = \operatorname{P}(K | C) &= \frac{\operatorname{P}(C | K ) \operatorname{P}( K )}{\operatorname{P}(C | K ) \operatorname{P}( K ) + \operatorname{P}(C | K ) \operatorname{P}( K )} \\ &= \frac{1( N / 20)}{1( N / 20) + 0.5(20 − N ) / 20} = \frac{N}{N + 0.5(20 − N )} \\ 0.824(0.5 N + 10) &= N \\ 8.24 &= 0.588 N \\ N &= 14. \end{align*} [[/math]]

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.