Chapter 6: Random Variables

Life insurance A life insurance company sells a term insurance policy to a$21$-year-old male that pays $100,000$if the insured dies within the next $5$years. The probability that a randomly chosen male will die each year can be found in mortality tables. The company collects a premium of $250$each year as payment for the insurance. The amount Y that the company earns on this policy is $250$per year, less the $100,000$that it must pay if the insured dies. Here is a partially completed table that shows information about risk of mortality and the values of $Y=$profit earned by the company:

(a) Copy the table onto your paper. Fill in the missing values of $Y$.

(b) Find the missing probability. Show your work.

(c) Calculate the mean $\mu Y$. Interpret this value in context.

Compute and interpret the standard deviation of$X.$

Knees Patients receiving artificial knees often experience pain after surgery. The pain is measured on a subjective scale with possible values of 1 (low) to 5 (high). Let X be the pain score for a randomly selected patient. The following table gives part of the probability distribution for X.

(a) Find $P(X=5)$

(b) If two patients who received artificial knees are chosen at random, what’s the probability that both of them report pain scores of $1$or $2$? Show your work.

(c) Compute the mean and standard deviation of $X$. Show your work.

Toss $4$times Suppose you toss a fair coin $4$times. Let $X=$the number of heads you get.

(a) Find the probability distribution of$X$.

(b) Make a histogram of the probability distribution. Describe what you see.

(c) Find $P(X\le 3)$ and interpret the result.

What is the probability that a randomly chosen subject completes at least $3$puzzles in the five-minute period while listening to soothing music?

(a) 0.3 (c) 0.6 (e) Cannot be determined

(b) 0.4 (d) 0.9

Fire insurance Suppose a homeowner spends $\$300$for a home insurance policy that will pay out $\$200,000$if the home is destroyed by fire. Let $Y=$the profit made by the company on a single policy. From previous data, the probability that a home in this area will be destroyed by fire is $0.0002$.

(a) Make a table that shows the probability distribution of Y.

(b) Compute the expected value of Y. Explain what this result means for the insurance company

A test for extrasensory perception (ESP) involves asking a person to tell which of $5$shapes—a circle, star, triangle, diamond, or heart—appears on a hidden computer screen. On each trial, the computer is equally likely to select any of the $5$shapes. Suppose researchers are testing a person who does not have ESP and so is just guessing on each trial. What is the probability that the person guesses the first $4$shapes incorrectly but gets the fifth correct?

a). $1/5$

b). ${\left(\frac{4}{5}\right)}^4$

c). ${\left(\frac{4}{5}\right)}^4\left(\frac{1}{5}\right)$

d). $\left(\frac{5}{1}\right){\left(\frac{4}{5}\right)}^4\left(\frac{1}{5}\right)$

e).$4/5$

Using Benford's law According to Benford's law (Exercise $5$, page $353$), the probability that the first digit of the amount of a randomly chosen invoice is an $8$or a $9$is $0.097$. Suppose you examine randomly selected invoices from a vendor until you find one whose amount begins with an $8$or a $9$.

(a) How many invoices do you expect to examine until you get one that begins with an $8$or $9$? Justify your answer.

(b) In fact, you don't get an amount starting with an $8$or $9$until the ${40}^{th}$invoice. Do you suspect that the invoice amounts are not genuine? Compute an appropriate probability to support your answer.

101. Job reads that 1 out of 4 eggs contains salmonella bacteria. So he never uses more than 3 eggs in cooking. If eggs do or don't contain salmonella independently of each other, the number of contaminated eggs when Joe uses 3 chosen at random has the following distribution:

(a) binomial; $n=4$and$p=1/4$

(b) binomial; $n=3$and$p=1/4$

(c) binomial; $n=3$and$p=1/3$

(d) geometric;$p=1/4$

(e) geometric;$p=1/3$

In the previous exercise, the probability that at least $1$of Joe's $3$eggs contains salmonella is about

(a) $0.84$.

(b) $0.68$.

(c) $0.58$.

(d) $0.42$.

(e) $0.30$.