In Exercises 9–20, use the data in the following table, which lists survey results from high school drivers at least 16 years of age (based on data from “Texting While Driving and Other Risky Motor Vehicle Behaviors Among U.S. High School Students,” by O’Malley, Shults, and Eaton, Pediatrics, Vol. 131, No. 6). Assume that subjects are randomly selected from those included in the table. Hint: Be very careful to read the question correctly.

Texting and Alcohol If three of the high school drivers are randomly selected from the 4720 subjects who did not text while driving, find the probability that all three drove when drinking.


a. Assume that the selections are made with replacement. Are the events independent?

Verified step by step guidance

1. Step 1: Identify the relevant data from the table. The problem focuses on high school drivers who did not text while driving. From the table, we see that 156 drivers drove when drinking alcohol, and 4564 drivers did not. The total number of drivers who did not text while driving is 4720.
2. Step 2: Calculate the probability of selecting one driver who drove when drinking alcohol. This is given by dividing the number of drivers who drove when drinking alcohol (156) by the total number of drivers who did not text while driving (4720). The formula is: $\frac{156}{4720}$
3. Step 3: Since the selections are made with replacement, the probability of selecting the same type of driver remains constant for each selection. To find the probability that all three drivers drove when drinking alcohol, multiply the probability of selecting one such driver by itself three times. The formula is: ${\frac{156}{4720}}^3$
4. Step 4: Determine whether the events are independent. In this case, because the selections are made with replacement, the outcome of one selection does not affect the outcome of another. Therefore, the events are independent.
5. Step 5: Conclude the process by noting that the probability calculation involves raising the single-event probability to the power of three, and the independence of events is confirmed by the 'with replacement' condition.

Key Concepts

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, it involves calculating the chance that all three randomly selected high school drivers who do not text while driving also drove after drinking alcohol. Understanding how to compute probabilities is essential for answering the question accurately.

Independent Events

Two events are considered independent if the occurrence of one does not affect the occurrence of the other. In this scenario, if selections are made with replacement, the probability of selecting a driver who drove after drinking remains constant across selections, indicating that the events are independent. This concept is crucial for determining how to calculate the overall probability of multiple events occurring.

Sampling with Replacement

Sampling with replacement means that after an individual is selected from a population, they are returned to the population before the next selection. This method ensures that each selection is made from the same total number of subjects, maintaining the same probabilities for each draw. Understanding this concept is vital for correctly applying probability rules in the given scenario.