Final answer:
These inquiries test a student's understanding of probability, random variables, and their distributions, involving identification of the right distributions, calculation of probabilities, and finding measures like mean and standard deviation.
Step-by-step explanation:
The overarching theme of these questions is probability and statistics, particularly focusing on the concept of random variables and their distributions. To assist the student:
Define the random variable X as the specific outcome of interest based on the context given in each scenario.
Identify the appropriate probability distribution to use. For countable outcomes, a binomial or Poisson distribution might be appropriate, depending on whether the number of trials or the rate of occurrence is known.
Calculate probabilities by applying the correct formulas based on the chosen distribution. This might involve using a binomial probability formula, a Poisson probability function, or standard deviation and mean formulas for a normal distribution.
Determine expected values, which in a probability context is often referred to as the mean of the distribution, and other measures like standard deviation to understand the variability of the distribution.
For calculating specific probabilities or expected values, numerical data from the problem must be plugged into the appropriate formulas. Without specific values provided in the question, a general guidance on the approach is given above.