Question

You are allowed to take a certain test three times, and your final score will be the maximum of the test scores. Your score in test i, where i = 1, 2, 3, takes one of the values from i to 63 with equal probability 1/(64-i), independently of the scores in other tests.

What is the probability that your final score is 46?

What is the probability that your final score is 61?

I need the exact numerical answer with these numbers. Thanks a lot.

Answer 1

By independence, we have

P(max{X1,X2,X3} ≤x) =P(X1≤x)P(X2≤x)P(X3≤x)

= x/63 * (x-1)/62* (x-2)/61

Notice that we can’t have x <3 for the maximum because the minimum of the third exam is 3. To compute the PMF at x, just take P(max{X1,X2,X3} ≤x)−P(max{X1,X2,X3}≤x−1)

What is the probability that your final score is 46?

P(max(X1,X2,X3) = 46) = F(46) - F(45)

= 46/63 * (46-1)/62* (46-2)/61 - 45/63 * (45-1)/62* (45-2)/61

= 0.0249301

What is the probability that your final score is 61?

=61/63 * (61-1)/62* (61-2)/61 - 60/63 * (60-1)/62* (60-2)/61

= 0.044572