- The local animal shelter adopts out cats and dogs each week with the probabilities shown below. a) Draw the probability distribution b) Find the mean, variance, and standard deviation for the number of animals adopted each week. c) What is the probability that the shelter finds homes for at least 6 animals in a given week? X 4 5 6 7 8 9 P(X) 0.08 0.18 0.22 0.15 0.16 0.21

X 4 5 6 7 8 9

PX 0.08 0.18 0.22 0.15 0.16 0.21

2) Based on a Harris poll, among adults who regret getting tattoos, 25% say that they were too young when they got their tattoos. Assume that 12 adults who regret getting tattoos are randomly selected.

- a) Why can x, the number of adults who say they were too young when they got their tattoos, be considered a binomial random variable?
- b) What is the probability that all of the adults say they were too young?
- c) What is the most likely number of adults who say they were too young when they got their tattoo?
- d) Which number(s) of adults who say they were too young are unlikely to be obtained in the selection?

3) An egg carton contains 24 eggs, 5 of which are cracked. We pick 8 eggs for boiling.

- a) What is the probability all the cracked eggs are selected?
- b) What is the probability that none of the cracked eggs are selected?
- c) What is the probability that less than 4 of the cracked eggs are selected?

4) The life span of oil-drilling bits depends on the types of rock and soil that the drill encounters, but it is estimated that the mean length of life is 75 hours. Suppose an oil exploration company purchases drill bits that have a life span that is approximately normally distributed, with a mean equal to 70 hours and a standard deviation equal to 14 hours.

- a) What proportion of the company’s drill bits will fail before 60 hours of use?
- b) What proportion will last at least 60 hours?
- c) What proportion will have to be replaced after more than 90 hours of use?
- d) After how many hours will 30% of drill bits have to be replaced?

5) Standing eye heights of women are normally distributed with a mean of 1564 mm and a standard deviation of 60 mm.

- a) What proportion of women have eye heights between 1475 mm and 1550 mm?
- b) A door peephole is placed at a height that is uncomfortable for women with standing eye heights greater than 1635 mm. What percentage of women will find that height uncomfortable? c) In selecting the height of a door peephole, the architect wants its height to be suitable for the highest 99% of standing eye of women. What standing height of women separates the highest 99% of standing eye heights from the lowest 1%?

6) A classic example of the Poisson distribution involves the number of deaths caused by horse kicks of men in the Prussian Army between 1875 and 1894. Data for 14 corps were combined for the 20-year period, and the 280 corps-year included a total of 196 deaths.

- a) Why is x, the number of deaths in a corps-year, is a Poisson random variable?
- b) After finding the mean number of deaths per corps-year, find the probability that a randomly selected corps-year has 0 death
- c) find the probability that a randomly selected corps-year has 2 deaths
- d) What is the maximum number of deaths that can be expected to occur for any given corps-year?

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