1.If a publisher of nontechnical books takes great pains to ensure that its books are free of typographical errors, so that the probability of any given page containing at least one such error is .005 and errors are independent from page to page, what is the probability that one of its 400-page novels will contain at most two pages with errors?
Compute the exact probability.
2.I throw two (fair and independent) dice, and observe the sum of the two numbers. I keep throwing until I see a sum of either 6, 7 or 8, then I stop. What is the probability that I will stop at a 7 ?
3.According to a survey, roughly 10% of US men and 90% of US women are cat lovers. Of US adults, 48% are men and 52% are women. Suppose now that we randomly select a US adult.
(a) What is the probability that the selected person is a man?
(b) If we already know that this selected adult is not a cat lover, what is the probability that the selected person is a man?
(c) Compare the two probabilities in (a) and (b), which probability is higher? Is that what you should expect a priori? Why?
4.a. Let X be a hypergeometric r.v. with parameters N = 300, M = 100, n = 3. Work out the p.m.f. of X.
b. Let Y be a binomial r.v. with parameters n = 3 and p = 1/3. Work out the p.m.f. of Y .
c. Calculate the expected value and variance of X.
d. What is the c.d.f. of Y ?
e. Hypergeometric distribution and binomial distribution are two different distri- butions, but if you worked out the p.m.f.'s of X and Y correctly, you should realize that the two p.m.f.'s are very close. Why is that ?
5.Suppose that the proportions of blood phenotypes in a particular population are as follows:
A - 0.42, B - 0.10, AB - 0.04, O - 0.44.
Assuming that the phenotypes of two randomly selected individuals are independent of one another. what is the probability that the phenotypes of two randomly selected individuals match ?