Distributions of random variables

GRE scores, Part II. In Exercise 4.3 we saw two distributions for GRE scores: $N(\mu=151,\sigma=7)$ for the verbal part of the exam and $N(\mu=153,\sigma=7.67)$ for the quantitative part. Use this information to compute each of the following:

1. The score of a student who scored in the $80^{th}$ percentile on the Quantitative Reasoning section.

2. The score of a student who scored worse than 70% of the test takers in the Verbal Reasoning section.

Is it Bernoulli? Determine if each trial can be considered an independent Bernoulli trial for the following situations.

1. Cards dealt in a hand of poker.

2. Outcome of each roll of a die.

Eye color, Part I. A husband and wife both have brown eyes but carry genes that make it possible for their children to have brown eyes (probability 0.75), blue eyes (0.125), or green eyes (0.125).

1. What is the probability the first blue-eyed child they have is their third child? Assume that the eye colors of the children are independent of each other.

2. On average, how many children would such a…

Bernoulli, the mean. Use the probability rules from Section an exercise given elsewhere in the book to derive the mean of a Bernoulli random variable, i.e. a random variable $X$ that takes value 1 with probability $p$ and value 0 with probability $1-p$. That is, compute the expected value of a generic Bernoulli random variable.

Underage drinking, Part I. Data collected by the Substance Abuse and Mental Health Services Administration (SAMSHA) suggests that 69.7% of 18-20 year olds consumed alcoholic beverages in any given year.

1. Suppose a random sample of ten 18-20 year olds is taken. Is the use of the binomial distribution appropriate for calculating the probability that exactly six consumed alcoholic beverages? Explain…

Underage drinking, Part II. We learned in Exercise 4.17 that about 70% of 18-20 year olds consumed alcoholic beverages in any given year. We now consider a random sample of fifty 18-20 year olds.

1. How many people would you expect to have consumed alcoholic beverages? And with what standard deviation?

2. Would you be surprised if there were 45 or more people who have consumed alcoholic beverages?

3. …

Eye color, Part II. Exercise 4.13 introduces a husband and wife with brown eyes who have 0.75 probability of having children with brown eyes, 0.125 probability of having children with blue eyes, and 0.125 probability of having children with green eyes.

1. What is the probability that their first child will have green eyes and the second will not?

2. What is the probability that exactly one of their…

Exploring permutations. The formula for the number of ways to arrange $n$ objects is $n!=n\times(n-1)\times\cdots\times 2\times 1$. This exercise walks you through the derivation of this formula for a couple of special cases.

A small company has five employees: Anna, Ben, Carl, Damian, and Eddy. There are five parking spots in a row at the company, none of which are assigned, and each day the employees pull into a random parking spot. That is…

Rolling a die. Calculate the following probabilities and indicate which probability distribution model is appropriate in each case. You roll a fair die 5 times. What is the probability of rolling

1. the first 6 on the fifth roll?

2. exactly three 6s?

3. the third 6 on the fifth roll?

Sampling at school. For a sociology class project you are asked to conduct a survey on 20 students at your school. You decide to stand outside of your dorm’s cafeteria and conduct the survey on a random sample of 20 students leaving the cafeteria after dinner one evening. Your dorm is comprised of 45% males and 55% females.

1. Which probability model is most appropriate for calculating the probabil…

Customers at a coffee shop. A coffee shop serves an average of 75 customers per hour during the morning rush.

1. Which distribution have we studied that is most appropriate for calculating the probability of a given number of customers arriving within one hour during this time of day?

2. What are the mean and the standard deviation of the number of customers this coffee shop serves in one hour durin…

How many cars show up? For Monday through Thursday when there isn’t a holiday, the average number of vehicles that visit a particular retailer between 2pm and 3pm each afternoon is 6.5, and the number of cars that show up on any given day follows a Poisson distribution.

1. What is the probability that exactly 5 cars will show up next Monday?

2. What is the probability that 0, 1, or 2 cars will show…

Roulette winnings. In the game of roulette, a wheel is spun and you place bets on where it will stop. One popular bet is that it will stop on a red slot; such a bet has an 18/38 chance of winning. If it stops on red, you double the money you bet. If not, you lose the money you bet. Suppose you play 3 times, each time with a $1 bet. Let Y represent the total amount won or lost. Write a probability…

University admissions. Suppose a university announced that it admitted 2,500 students for the following year’s freshman class. However, the university has dorm room spots for only 1,786 freshman students. If there is a 70% chance that an admitted student will decide to accept the offer and attend this university, what is the approximate probability that the university will not have enough dormitor…

Auto insurance premiums. Suppose a newspaper article states that the distribution of auto insurance premiums for residents of California is approximately normal with a mean of $1,650. The article also states that 25% of California residents pay more than $1,800.

1. What is the Z-score that corresponds to the top 25% (or the $75^{th}$ percentile) of the standard normal distribution?

2. What is the mean insur…

Married women. The American Community Survey estimates that 47.1% of women ages 15 years and over are married.

1. We randomly select three women between these ages. What is the probability that the third woman selected is the only one who is married?

2. What is the probability that all three randomly selected women are married?

3. On average, how many women would you expect to sample before selecting…

Overweight baggage. Suppose weights of the checked baggage of airline passengers follow a nearly normal distribution with mean 45 pounds and standard deviation 3.2 pounds. Most airlines charge a fee for baggage that weigh in excess of 50 pounds. Determine what percent of airline passengers incur this fee.

Heights of 10 year olds, Part III. Heights of 10 year olds, regardless of gender, closely follow a normal distribution with mean 55 inches and standard deviation 6 inches.

1. What fraction of 10 year olds are taller than 76 inches?

2. If there are 2,000 10 year olds entering Six Flags Magic Mountain in a single day, then compute the expected number of 10 year olds who are at least 76 inches tall. (Y…