Problem 1
The probability that a subject in a clinical trial will experience an adverse reaction is thought to be (1/25) = 0.04. What is the probability that of the next 10 subjects,
| a) | Exactly two will experience an adverse reaction; |
| b) | No more than one adverse reaction will occur; |
| c) | No adverse reaction will occur; |
| d) | At least two adverse reactions will occur; |
| e) | At least two but no more than four adverse reactions will occur? |
Problem 2
One of the first published uses of the Poisson distribution was an 1898 study of accidental deaths due to kicks from horses, from twenty years' data on Prussian cavalry units. The data are very well fit by a Poisson distribution with a mean of m=0.7 annual deaths per army unit. For all three parts of this question you may assume that this Poisson distribution is exactly correct.
a) What is the probability the First Army Unit would experience zero deaths in a particular year, if the number X of deaths follows this distribution? Give both an exact mathematical expression and also a numerical answer accurate to three decimal places for Pr[X=0].
b) The data set includes annual death counts attributed to horse-kicks for 20 years on each of 14 Prussian units. About how many of those 280 annual unit records should report zero annual deaths? Give a numerical answer for the number N of unit-year's with ZERO deaths.
c) About how many deaths in all would you expect in this data, covering the 280 unit-years? Give a numerical answer for the total number D of deaths over the twenty years.
Problem 3
A young boy randomly places small firecrackers into the tips of five cigars in his father's cigar box, which contains 25 cigars. Unaware of his son's prank the father randomly selects three cigars and gives them to his boss. What is the probability that the father will not get fired? (Assume he will get fired if any of the three cigars contains a firecracker).
Problem 4
Problem 4.27, on page 166 of the textbook (Mendenhall & Sincich).
Optional Exercise
The random variables X and Y are independent, both with the Binomial distribution with N=2 and p=0.5; Z is independent of them both, with the Poisson distribution with mean m=1. Which has higher probability, the event [X=Y] or the event [X=Z]? Show your work.