Please give answer and show your work for all questions

Question

Let’s assume the following statements are true: Historically, 75% of the five-star football recruits in the nation go to universities in the three most competitive athletic conferences. Historically, five-star recruits get full football scholarships 93% of the time, regardless of which conference they go to. If this pattern holds true for this year’s recruiting class, answer the following:

a. Based on these numbers, what is the probability that a randomly selected five-star recruit who chooses one of the best three conferences will be offered a full football scholarship?

b. What are the odds a randomly selected five-star recruit will not select a university from one of the three best conferences? Explain.

c. Explain whether these are independent or dependent events. Are they Inclusive or exclusive? Explain.

Answer 1

a. The probability that a randomly selected five-star recruit who chooses one of the best three conferences will be offered a full football scholarship:

Since the scholarships are offered to recruits regardless of the conference they choose, the probability remains 93%. The statistics about how often they go to the three most competitive conferences doesn't change this.

b. The odds a randomly selected five-star recruit will not select a university from one of the three best conferences:

Given that 75% of the five-star football recruits go to universities in the three most competitive athletic conferences, there is a 25% (100% - 75%) chance that a recruit will not select a university from one of the three most competitive conferences.

c. The two events – the recruit going to one of the top three conferences and the recruit getting a full scholarship – are independent. This is because the probability of a recruit getting a scholarship (93%) does not depend on which conference the recruit chooses to go to. In other words, the outcome of one event does not affect the probability of the other event.

Inclusive events are those which can happen at the same time. A five-star recruit choosing to go to a top conference and also getting a full scholarship are inclusive events, as one recruit can experience both. On the other hand, exclusive events are those which cannot happen at the same time. If we consider the event of a recruit going to a top conference and the event of a recruit not going to a top conference, these are exclusive, as one recruit cannot do both.

Answer 2

The answers are: To (a): 93%. Since the probability given is stated to be "regardless of which conference they choose," the chance that the five-star recruit will be offered a full football scholarship is not affected by their choice of whether or not to attend one of the three most competitive conferences.

To (b): 25%. SIne the probability that the five-star recruit will choose one of the three competitive conferences is 75%, the probability that he will not do so is 1 - 75%, or 25%.

To (c): Yes, these are independent events, and no, they are not exclusive events. Independent events are those for which knowing the outcome of either does not affect the probability of the other's happening. Exclusive events are those which cannot happen at the same time, and exclusive events can never be independent. For example, rolling a 4 on a standard die and rolling a 6 on that same throw of the die are exclusive events: You cannot roll a 4 and a 6 at the same time. If we know that we have rolled a 4, then the chance that we rolled a 6 at the same time drops from 1/6 to zero, and if we know that we have not rolled a 4, the chance that we have rolled a 6 at the same time increases from 1/6 to 1/5. If rolling a 4 and rolling a 6 were independent events, then the chance of rolling a 6 would be unaffected by our knowledge of whether or not we have rolled a 4.