Saturday, October 10, 2009

Lazy student

A student guesses randomly on a 50-question True/False test. Find the probability that the number of correct guesses is divisible by 4.

1 comment:

  1. The answer is 1/4.

    Once the student has answered the other questions, the parity of his score (odd or even) will depend on his response to the last question. Therefore, his score is even with probability 1/2.

    Suppose that the score S is even. If he had reversed all of his responses, writing true for false and false for true, then his score would have been (50 - S), which is also an even number. But note that exactly one of the two quantities S and (50 - S) is a multiple of 4. So, if S is even, then the probability is 1/2 that it is also divisible by 4.

    Therefore, the probability that the score is divisible by 4 is 1/2 * 1/2 = 1/4.

    REMARK: This argument is valid when the number of questions is congruent to 2 modulo 4. In other cases, the probability would be slightly different than 1/4. It is a challenge to find the exact probability in those cases.