1. Consider the situation when a couple has 3 children. Assume that boys and girls are equally likely and the gender of any child is not influenced by the gender of any other child.

1. (a) Among three children, there is exactly one girl.
2. (b) Among three children, there are exactly two girls.
3. (c) Among three children, all are girls.

2. The principle of redundancy is used when the system reliability is improved through redundant or backup components. Assume that your alarm clock has a 0.975 probability of working on any given morning.

(a) What is the probability that your alarm clock will not work on the morning of an important final exam?

(b) If you have two such alarm clocks, what is the probability that they both fail on the morning of an important final exam?

(c) With one alarm clock, you have a 0.975 probability of being awakened. What is the probability of being awakened if you use two alarm clocks?

(d) Does second alarm clock result in greatly improved reliability?

3. With one method of the procedure called acceptance sampling, a sample of items is randomly selected without the replacement, and the entire batch is rejected if there is at least one defect. The Medtyme Company has just manufactured 5000 blood pressure monitors, and 4% are defective. If 3 of them are selected and tested, what is the probability that the entire batch will be rejected?

4. A study of hunting injuries and he wearing of hunter orange clothing showed that among 123 hunters injured when mistaken for game, 6 were wearing orange. If a follow up study begins with the random selection of hunters from this sample of 123, find the probability that the first two selected hunters were both wearing orange.

1. Assume that the first hunter is replaced before the next one is selected.
2. Assume the first hunter is not replaced before the second one is selected.
3. Given a choice between selecting with replacement and selecting without

replacement, which choice makes more sense in this situation? Why?

5. A cartoon contains 12 eggs, 3 of which are cracked. If we randomly select 5 of the eggs for hard boiling, what is the probability of the following events?

1. All of the cracked eggs are selected.
2. None of the cracked eggs are selected.
3. Two of the cracked eggs are selected.

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