MathStat474 - HW 4

Posted: Thu Feb 5. Due: Thu Feb 12.

Notation: Please notice that our textbook uses the notation \(A'\) for the complement of the event \(A\). So when it writes \(P(A')\), they mean \(P(\mbox{not }A)\).

Problem 1.

According to www.jointogether.com, in 2000, 87% of all suicides were committed by males, 56% of all suicides were committed using a gun, and 10% of all suicides were committed by women not using a gun. (We’ll use M for suicide committed by a male, F[= not M] for suicide committed by female, and G for a suicide committed using a gun.) Which of the following is a correct representation of the given information in a two-way probability table?

Problem 2.

Researchers studied thousands of court cases. For each case, they recorded the jury’s decision. In addition, they asked the judge in each case how he or she would have decided the same case if there were no jury. In 67% of the cases the jury voted to convict, in 83% of the cases the judge would have convicted, and in 19% of the cases only the judge would have convicted. Let A be the event “jury convicts” Let B be the event “judge convicts”.

Complete the two-way probability table of the same format as in the previous problem!

Problem 3.

Solve exercise 3.11.95 in the textbook. (Conditional probability, law of total probability.)

A batch of 500 containers of frozen orange juice contains 5 that are defective. Two are selected, at random, without replacement, from the batch. Let \(A\) and \(B\) denote the events that the first and second containers selected are defective, respectively.

Are \(A\) and \(B\) independent events?
If the sampling were done with replacement, would \(A\) and \(B\) be independent?

Problem 4.

Solve exercise 3.11.109 in the textbook. (Conditional probability, law of total probability, and Bayes.)

Customers are used to evaluate preliminary product designs. In the past, 95% of highly successful products received good reviews, 60% of moderately successful products received good reviews, and 10% of poor products received good reviews. In addition, 40% of products have been highly successful, 35% have been moderately successful, and 25% have been poor products.

What is the probability that a product attains a good review?
If a new design attains a good review, what is the probability that it will be a highly successful product?
If a product does not attain a good review, what is the probability that it will be a highly successful product?

Problem 5.

[This is not a required problem. Doing it will earn you up to 5 points of extra credit.]

Prove that \[ P(A'\cap B′)=1+P(A\cap B)-P(A)-P(B). \]

Typing up work

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