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STAT7055 Semester 2 2022

Topic 2 Tutorial Questions

1. (Keller Ex 6.91 (10e) or 6.113 (11e)) In a four-cylinder engine there are four spark plugs. If any one of them malfunctions, the car will idle roughly, and power will be lost. Suppose that for a certain brand of spark plugs, the probability that a spark plug will function properly after 5000 miles is 0.9. Assuming that the spark plugs operate independently, what is the probability that the car will idle roughly after 5000 miles?

2. (Keller Ex 6.89 & 6.90 (10e) or 6.111 & 6.112 (11e)) The eﬀect of an antidepressant drug varies from person to person. Suppose that the drug is eﬀective on 80% of women and 65% of men. It is known that 66% of the people who take the drug are women.

(a) What is the probability that the drug is eﬀective?

(b) Suppose you are told that the drug is eﬀective. What is the probability that the

drug taker is a man?

3. (Keller Ex 6.94 (10e) or 6.116 (11e)) In Canada, criminals are entitled to parole after serving only one-third of their sentence. Virtually all prisoners, with several exceptions including murderers, are released after serving two-thirds of their sentence. The govern- ment has proposed a new law that would create a special category of inmates based on whether they had committed crimes involving violence or drugs. Such criminals would be subject to additional detention if the Correction Services judge them highly likely to re-oﬀend. Currently, 27% of prisoners who are released commit another crime within

2 years of release. Among those who have re-oﬀended, 41% would have been detained under the new law, where 31% of those who have not re-oﬀended would have been detained.

(a) What is the probability that a prisoner who would have been detained under the

new law does commit another crime within 2 years?

(b) What is the probability that a prisoner who would not have been detained under

the new law does commit another crime within 2 years?

4. Suppose I have four fair coins, two which have a head on both sides, one which has a tail on both sides, and one which is normal (a head on one side and a tail on the other side).

(a) I shut my eyes, pick one of the four coins at random, and ﬂip it. What is the

probability that the lower face of the coin is a head?

(b) I open my eyes and see that the coin is showing heads (on the upper face). What

is the probability that the lower face is a head?

5. (Topic 2 Example 4) There are three buckets that weigh 1kg, 2kg and 3kg, respectively. A bucket weighing ikg contains i white balls and 5 − i black balls, for i = 1, 2, 3. For example, the bucket weighing 1kg contains 1 white ball and 4 black balls. Suppose a bucket is chosen with probability proportional to its weight and two balls are randomly selected (without replacement) from this bucket.

(a) Find the probability that both balls selected are white.

(b) If both balls selected are white, what is the probability that the bucket weighing

3kg was chosen?

6. Discussion Question

Monty Hall problem. You are on a game show where you are given the choice of three suitcases, only one of which contains a million dollars. You choose a suitcase, but before you open it, the game show host opens one of the other two suitcases and shows you that it’s empty (we can assume the host knows which suitcase contains the million dollars). You are now given a choice to either stay with the suitcase you originally chose, or to switch to the other unopened suitcase. What should you do? What happens if you were instead given a choice of four suitcases, only one of which contains a million dollars? Back in the ﬁrst situation with three suitcases, what happens if, after you choose a suitcase, this time the host chooses one of the other two suitcases randomly and shows you that it’s empty?

7. swirl

Work through lessons 3 and 4 of the R Programming course.