SOLUTION: 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 confe

Question 1163922: 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 by CPhill(2264) (Show Source): You can put this solution on YOUR website!
Based on the statistics provided, here are the calculations and logical breakdowns for this year's recruiting class:
### a. Probability of a Scholarship in a Top Conference
The probability is **93%** (or **0.93**).
**Reasoning:** The prompt states that five-star recruits get full scholarships 93% of the time **"regardless of which conference they go to."** This implies that the scholarship rate and the conference choice are treated as independent factors in this dataset. Therefore, knowing the recruit chose a top conference does not change the established 93% probability.
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### b. Odds of Not Selecting a Top Conference
The odds are **1 to 3** (or $1:3$).
**Explanation:**
1. **Probability ($P$):** We know 75% ($3/4$) of recruits choose a top conference. Therefore, the probability of **not** choosing one is 25% ($1/4$).
2. **Calculating Odds:** Odds are expressed as the ratio of *Successes* to *Failures* (or in this case, *Not Selecting* vs. *Selecting*).
* $\text{Odds} = \frac{P(\text{Not Top})}{P(\text{Top})} = \frac{25\%}{75\%} = \frac{1}{3}$
3. **Interpretation:** For every 1 recruit who chooses a school outside the top three conferences, there are 3 recruits who choose a school within them.
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### c. Event Relationships
**Independent vs. Dependent**
These are **Independent** events.
* **Why?** The prompt explicitly states the scholarship rate is 93% "regardless of which conference they go to." This means the outcome of the first event (choosing a conference) has no effect on the probability of the second event (getting a scholarship).
**Inclusive vs. Exclusive**
These are **Inclusive** events.
* **Why?** Mutually exclusive events are things that cannot happen at the same time (like a coin landing on both Heads and Tails). In this scenario, a recruit **can** both choose a top conference **and** receive a full scholarship simultaneously. Since these two outcomes can overlap in a single person, they are inclusive.
***
Do you have a specific recruit's data you're trying to model, or are you looking for the combined probability of both events happening at once?
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