MathStat474 - HW 4

Posted: Wed Sep 10. Due: Wed Sep 17.

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 2.58 in the textbook. (Compute some probabilities when rolling a pair of dice.)

A pair of fair dice is tossed. Find the probability of getting:

  1. a total of 8;
  2. at most a total of 5.

Problem 4.

Solve exercise 2.72 in the textbook. (Prove a certain equality of probabilities!)

Prove that

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

Problem 5.

Solve exercise 2.74 in the textbook:

A class in advanced physics is composed of 10 juniors, 30 seniors, and 10 graduate students. The final grades show that 3 of the juniors, 10 of the seniors, and 5 of the graduate students received an A for the course. If a student is chosen at random from this class and is found to have earned an A, what is the probability that he or she is a senior?

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