Final answer:
We calculated initial and conditional probabilities for Renate's chance to win a license from the U.S. government's lottery, determined the dependency of events, and identified binomial probability situations. We also discussed analyzing probabilities with conditions in driver behavior.
Step-by-step explanation:
The question pertains to understanding how to calculate probabilities, conditional probabilities, and distinguishing between independent, dependent, and mutually exclusive events. In the context of the U.S. government's lottery for licenses, Renate's initial chance of winning was 55,000 out of 6.5 million, simplified to the probability statement P(G) = 55,000 / 6,500,000. When narrowed down to 110,000 finalists, Renate's conditional probability of winning was 55,000 out of 110,000, represented by P(G|F) = 55,000 / 110,000. These probabilities help us understand that G and F are dependent events because Renate's chances of winning the license change after becoming a finalist, which should be reflected numerically by comparing initial and conditional probabilities. They are not mutually exclusive because being a finalist does not preclude one from also being a license winner.
For the driver's license age data, if we assume a normal distribution, we can use the mean of the collected ages from each region to determine whether teenagers across the country obtain their licenses at approximately the same age. For exercise four, identifying a situation as a binomial problem involves checking for two outcomes (pass or fail), a fixed number of trials (50 people), and independent trials with the same probability of success, which in this case is the 65% pass rate on the first try.
Finally, when analyzing probabilities involving conditions, such as using a cell phone while driving and having no traffic violations, we calculate the probability of both conditions occurring together to better understand the relationship between these factors in driver behavior.