Final answer:
The question requires applying the principle of inclusion-exclusion to find the probabilities of certain behaviors among a class of 500 college students. The probabilities are based on known quantities of students engaging in behaviors such as smoking, drinking, and eating between meals, accounting for overlap between these behaviors.
Step-by-step explanation:
The student's question asks for the probability of certain behaviors among a college class of 500 students. We need to use the principle of inclusion-exclusion to find the individual probabilities for the scenarios given. This principle accounts for the overlaps between behaviors (smoking, drinking, eating between meals) and ensures that no student is counted more than once in our calculations.
To find the probability that a student smokes but does not drink alcoholic beverages, we take the total number of students who smoke (190) and subtract those who both smoke and drink (112), as well as those who engage in all three activities (38). The remaining number of students only smoke. Then we divide this by the total number of students to get the probability.
To calculate the probability that a student eats between meals and drinks alcoholic beverages but does not smoke, we consider those who eat between meals and drink (65), subtract those who also smoke (38), and divide by the total number of students.
Finally, to determine the probability that a student neither smokes nor eats between meals, we need to consider all those who do neither and divide by the total student population. This involves subtracting from the total number (500), the number of students who smoke (190), eat between meals (196), and those who do both (81), adding back those who engage in all three activities (38), as they have been subtracted twice.