We know that 5% of the people in a certain population have a virus. Suppose that I draw a random sample of 100 individuals: the population is so large in the order of millions) so that, even though I perform the sampling without replacement, my samples may be considered independent of one another (that is, (i) the first and second individuals having the virus are independent events, and (ii) regardless of the first individual, the probability of picking another individual with the virus is still 5%). Let N be the random variable describing the number of individuals, in my sample, with the virus. (a) What is the probability distribution of N? (Do you recognize it? Does it have a special name?)
(b) Compute the expected number E(N) of individuals, in the sample, with the virus.
(c) What is the prob. of getting exactly this many individuals with the virus, in our sample of 100 individuals?
(d) Compute the standard deviation of N. Hint: Use the appropriate formula...

Aprisoner is trapped in a cell containing three doors. the first door leads to a tunnel that returns him to his cell after two days of travel. the second leads to a tunnel that returns him to his cell after three days of travel. the third door leads immediately to freedom. (a) assuming that the prisoner will always select doors 1, 2 and 3 with probabili- ties 0.5,0.3,0.2 (respectively), what is the expected number of days until he reaches freedom? (b) assuming that the prisoner is always equally likely to choose among those doors that he has not used, what is the expected number of days until he reaches freedom? (in this version, if the prisoner initially tries door 1, for example, then when he returns to the cell, he will now select only from doors 2 and 3.) (c) for parts (a) and (b), find the variance of the number of days until the prisoner reaches freedom. hint for part (b): define ni to be the number of additional days the prisoner spends after initially choosing door i and returning to his cell.