In an informal conversation the secretary of the Department

In an informal conversation, the secretary of the Department of Bayesian Engineering told three graduate students (A, B, and C) who had just taken their Ph.D. qualifying exam that one of them did not pass. The secretary, however, was under strict order from the examination committee not to inform any student of his or her results yet. Mr. A, who had studied very hard, had become a knowledgeable Bayesian, and could not waltz he asked the secretary if she could at least tell him whether ?B? passed or not. He felt comfortable asking this question since he was not requesting information about his own test result, and given the fact that they were told that only one student failed the test, it was obvious that either \'\'B\'\' or \'\'C\'\' must have passed. The secretary, after some deliberations, decided that providing such information would not be against her instructions, since she would not be telling ?A? anything about his results. She then told \'\'A\'\' that ?B? had actually passed the test. Was the secretary correct in her assessment that she was not providing any information about A\'s examination result? Provide quantitative reason. [20 points]

Solution

Since there are 3 graduate students and it is known that one of them fails while the other two pass

The probability that A failsP(AF)= 1/3 and probability that A passes is P(AP)= 2/3

The probability that B failsP(BF)= 1/3 and probability that B passes is P(BP)= 2/3

The probability that C failsP(CF)= 1/3 and probability that C passes is P(CP)= 2/3

When no result was disclosed by the Secretary the possible ways in which one of them fails while the other two pass are:

Probability that A failed and B and C passed=1/3 X 2/3 X 2/3=4/27.............................1

Probability that B failed and A and C passed=2/3 X 1/3 X 2/3=4/27..............................2

Probability that C failed and A and C passed=2/3 X 2/3 X 1/3=4/27.............................3

The probability that A passesh equals=2 +3= 4/27 +4/27=8/27=0.2963

If however, the secretary discloses the results of B to A the probabilities change;

The probability that A failsP(AF)= 1/2 and probability that A passes is P(AP)= 1/2

The probability that C failsP(AF)= 1/2 and probability that C passes is P(AP)= 1/2

So now A asseses that the ways in which the results would take shape could be as follows:

A fails P(AP)=1/2 and B passes P(BP)=1/2= 1/2 X 1/2=1/4

B fails P(BF)=1/2 and A passes P(AP)=1/2=1/2 X 1/2= 1/4

The sum of the two probabilities yield the probability that would reflect the possible results=1/2 and that A passes is 1/4

Thus, though the Secretary does not reveal to A his result, the result of B decreased the probability of his passing from 0.2963 to 0.25

Hence, the Secretary has thus provided some information contribution to the examination result.