Question

A student answers a multiple choice question with ? choices in the
following manner:
The probability that the student studied the specific subject is ?, thus,
he knows the answer. Otherwise, he guesses.
a) What is the probability that the student studied a subject given that
he answered correctly?
b) Calculate this probability for the cases ? = 1 and ? → ∞. Explain
the results.

Answer 1

C : answer is correct

S : studied subject

given

P(S) = p

P(C | S) = 1

P(not S) = 1-p

P(C | not S) = 1/no. of options = 1/m

a.

P(C) = P(C|S)*P(S) + P(C|not S)*P(not S)

= 1*p + (1/m)*(1-p)

= (m*p + 1-p)/m

= (1 + (m-1)*p) / m

P(S | C) = P(C|S)*P(S) / P(C)

= 1*p / ((1 + (m-1)*p) / m)

= m*p / (1+(m-1)*p)

b.

for m = 1

P(S|C) = m*p / (1+(m-1)*p)

= 1*p / (1+(1-1)*p)

= 1*p / (1)

= p

for m = 1 : P(S|C) = p

for m → ∞

P(S|C) = m*p / (1+(m-1)*p)

= p / ( (1+(m-1)*p) / m)

= p / (1/m + (1 - (1/m))*p)

(for m → ∞ : 1/m → 0)

(put 1/m = 0)

=p / (0 + (1 - 0)*p)

= p / (p)

= 1

for m → ∞ : P(S|C) = 1