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MAT1830 - Discrete Mathematics for Computer Science

Assignment #4

(1)    (i) Out of all the visitors to a website, 25% are bots and the rest are humans. The website tries to determine whether a given visitor is a bot or a human by asking them three questions. For each of these question, humans get the answer right 80% of the time and bots get it right 20% of the time (this is independent of the other questions they’re asked). What is the probability that a visitor to the website (selected uniformly at random) is a bot, given that they get all three answers wrong?

(ii) Another website with a different visitor profile asks its visitors two questions (the questions work as described in (i)). Given that a visitor to this website (selected uniformly at random) answers both questions wrong, there is an exactly 75% probability that they are a bot. What fraction of this website’s visitors are bots?

(2) Let X and Y be independent random variables with distributions

Let W = 8X − 6Y and Z = XY .

(i) What is E[X]?

(ii) What is Var[X]?

(iii) What is E[W]?

(iv) Write down a table giving the probability distribution for Z .

(3) Of the 66 subsets of {1, 2, . . . , 12} with exactly 2 elements, one is chosen uniformly at random. Let X be lesser element of this subset and let Y be sum of the elements of this subset.

(i) Are the events“X = 1”and“Y = 13”independent?

(ii) Are the random variables X and Y independent?

(iii) What is Pr(Y ≥ 17 | X ≤ 7)?