This problem involves conditional probability, specifically calculating the proportion of capable college students among those admitted. We will use Bayes' Theorem to solve it.

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Step 1: Define the Events

• $C$: Candidate is capable.
• $\overline{C}$: Candidate is incapable.
• $P$: Candidate passes the test.

Given probabilities:

1. $P(C) = 0.4$
2. $P(\overline{C}) = 0.6$ (since $P(C) + P(\overline{C}) = 1$)
3. $P(P | C) = 0.8$ (probability of passing if capable)
4. $P(P | \overline{C}) = 0.25$ (probability of passing if incapable)

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Step 2: Calculate the Total Probability of Passing ($P(P)$)

Using the law of total probability:

$P(P) = P(P | C)P(C) + P(P | \overline{C})P(\overline{C})$

Substitute the given values:

$P(P) = (0.8)(0.4) + (0.25)(0.6)$

$P(P) = 0.32 + 0.15 = 0.47$

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Step 3: Find the Proportion of Capable Students Among Those Who Passed ($P(C | P)$)

Using Bayes' Theorem:

$P(C | P) = \frac{P(P | C)P(C)}{P(P)}$

Substitute the known values:

$P(C | P) = \frac{(0.8)(0.4)}{0.47}$

$P(C | P) = \frac{0.32}{0.47} \approx 0.6809$

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Final Answer:

The proportion of capable college students among those who passed is approximately:

$\boxed{68.09\%}$

Would you like further details or examples?

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Related Questions:

1. What is the probability that a candidate who failed the test is capable?
2. How does the proportion of capable students change if 50% of candidates are capable?
3. What would happen to the total probability of passing if all candidates were equally capable?
4. How is Bayes' Theorem applied in other real-world scenarios like medical testing or spam filtering?
5. What is the impact of increasing the test difficulty on the admission of capable students?

Tip:

Bayes' Theorem is invaluable for problems involving conditional probabilities, especially when distinguishing between true and false positives in real-world applications.