MAT1830 - Discrete Mathematics for Computer Science Assignment #4
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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)?
2023-05-30