Mensa is a high IQ society that admits people as members if theycan score at the 98th percentile or above on certain standardizedIQ (intelligence quotient) tests. On one such test, the StanfordBinet, the qualifying score is 132. The test consists of nquestions, each with m choices.
a) On any given test question, the person taking the test knowsthe answer with probability p. Assume that when the person does notknow the answer, the person guesses an option completely at random.Calculate the probability a person knew the answer to a question,given that they answered it correctly.
b) If a person receives a score of 132 or higher on the test,they are considered to have an IQ of 132 or higher. However,individuals with IQ less than 132 can also receive such scoresabout 0.1% of the time due to lucky guessing. Given that a personis labeled as having IQ of 132 or higher, what is the probabilitythey actually have IQ below 132? Assume that all individuals withIQ of 132 or higher receive an accurate score 95% of the time.
