According to the Agency for Healthcare Research and Quality, there were $409,706$ hospital visits for asthma-related illnesses in a recent year. The age distribution was as follows.

$$\begin{array}{cr} \hline \text{Age Range} & \text{Number}\\ \hline \text{Less than 1 year} & 7,866\\ 1- 17& 103,040\\ 18 - 44 & 79,659\\ 45-64 & 121,728\\ 65 - 84 & 80,649\\ \text{85 and up} & 16,764\\ \hline \text{Total} & 409,706\\ \end{array}$$

$\textbf{a.}\hspace{10pt}$ What is the probability that an asthma patient is between $18$ and $44$ years old?

$\textbf{b.}\hspace{10pt}$ What is the probability that an asthma patient is $65$ or older?

$\textbf{c.}\hspace{10pt}$ Using a cutoff of $0.05$, is it unusual for an asthma patient to be less than $1$ year old?

A section of an exam contains four True-False questions. A completed exam paper is selected at random, and the four answers are recorded. a. List all 16 outcomes in the sample space. b. Assuming the outcomes to be equally likely, find the probability that all the answers are the same. c. Assuming the outcomes to be equally likely, find the probability that exactly one of the four answers is “True.” d. Assuming the outcomes to be equally likely, find the probability that at most one of the four answers is “True.”

A certain ice cream parlor offers $15$ flavors of ice cream. You want an ice cream cone with three scoops of ice cream, all different flavors.

$\textbf{a.}\hspace{10pt}$ In how many ways can you choose a cone if it matters which flavor is on the top, which is in the middle, and which is on the bottom?

$\textbf{b.}\hspace{10pt}$ In how many ways can you choose a cone if the order of the flavors doesn’t matter?

The General Social Survey asked 46,349 women how many children they had. The results were as follows.

$$\begin{array}{cr} \hline \text{Number of Children} & \text{Number of women}\\ \hline 0 & 12,656\hspace{20pt}\\ 1 & 7,438\hspace{20pt}\\ 2 & 11,290\hspace{20pt}\\ 3 & 7,143\hspace{20pt}\\ 4 & 3,797\hspace{20pt}\\ 5 & 1,811\hspace{20pt}\\ 6 & 916\hspace{20pt}\\ 7 &522\hspace{20pt}\\ \text{8 or more} & 776\hspace{20pt}\\ \hline \text{Total} & 46,349\hspace{20pt}\\ \end{array}$$

$\textbf{a.}\hspace{10pt}$ What is the probability that a sampled woman has two children?

$\textbf{b.}\hspace{10pt}$ What is the probability that a sampled woman has fewer than three children?

$\textbf{c.}\hspace{10pt}$ Assume this is a simple random sample of U.S. women. Use the Empirical Method to estimate the probability that a U.S. woman has more than five children.

$\textbf{d.}\hspace{10pt}$ Using a cutoff of $0.05$, is it unusual for a woman to have no children?

A lot of 1000 components contains 300 that are defective. Two components are drawn at random and tested. Let A be the event that the first component drawn is defective, and let B be the event that the second component drawn is defective. a. Find P(A). b. Find P(B|A). c. Find $P(A \cap B).$ d. Find $P\left(A^{c} \cap B\right)$ e. Find P(B). f. Find P(A|B). g. Are A and B independent? Is it reasonable to treat A and B as though they were independent? Explain.

The monthly rents for apartments in a certain town have a mean of $\$800$ with a standard deviation of $\$150$. What can you determine about these data by using Chebyshev’s Inequality with $K = 3$?

A machine makes rivets that are used in the manufacture of airplanes. To be acceptable, the length of a rivet must be between $0.9$ centimeter and $1.1$ centimeters. The mean length of a rivet is $1.0$ centimeter, with a standard deviation of $0.05$ centimeter. What is the maximum possible percentage of rivets that are unacceptable?

In a survey of $500$ likely voters in a certain city, $275$ said that they planned to vote to reelect the incumbent mayor.

$\textbf{a.}\hspace{10pt}$ What is the probability that a surveyed voter plans to vote to reelect the mayor?

$\textbf{b.}\hspace{10pt}$ Interpret this probability by estimating the percentage of all voters in the city who plan to vote to reelect the mayor.

According to a survey by Nickelodeon TV, $88\%$ of children under $13$ in Germany recognized a picture of the cartoon character SpongeBob SquarePants. What is the probability that a randomly chosen German child recognizes SpongeBob?

A quiz consists of three true–false questions and two multiple-choice questions with five choices each. How many different sets of answers are there?

A manufacturer of cell phone batteries determines that the average length of talk time for one of its batteries is $470$ minutes. Suppose that the standard deviation is known to be $32$ minutes and that the data are approximately bell-shaped. Estimate the percentage of batteries that have z-scores between $−1$ and $1$.

Assume that a student is chosen at random from a class. Determine whether the events $A$ and $B$ are independent, mutually exclusive, or neither.

A: The student is a freshman.

B: The student is a sophomore

Determine whether the association between the two variables is positive or negative.

Weekly ice cream sales and weekly average temperature

A version of the dice game “craps” is played in the following manner. A player starts by rolling two balanced dice. If the roll (the sum of the two numbers showing on the dice) results in a 7 or 11, the player wins. If the roll results in a 2 or a 3 (called craps ), the player loses. For any other roll outcome, the player continues to throw the dice until the original roll outcome recurs (in which case the player wins) or until a 7 occurs (in which case the player loses). What is the probability that a player wins the game on the first roll of the dice?