In California, 29% of college students that earn a bachelor's degree have a starting salary of $35,000 or higher. Of starting entry positions, 40% have a salary of $35,000 or higher. What is the probability that a person in one of these positions has a bachelor's degree?
• 11.0%
• 13.8%
• 69.0%
72.5%
To find the probability that a person in one of these positions has a bachelor's degree, we can use Bayes' Theorem.
Let A be the event that a person has a starting salary of $35,000 or higher, and let B be the event that a person has a bachelor's degree.
We are given P(B) = 29% = 0.29, P(A|B) = 40% = 0.40, and P(A|B') = 0% (assuming the starting salary of $35,000 or higher is only for those with a bachelor's degree).
We want to find P(B|A), the probability that a person with a starting salary of $35,000 or higher has a bachelor's degree.
Using Bayes' Theorem:
\[P(B|A) = \frac{P(A|B) \cdot P(B)}{P(A)}\]
\[P(B|A) = \frac{0.40 \cdot 0.29}{P(A)}\]
\[P(B|A) = \frac{0.116}{P(A)}\]
We can find P(A) using the Law of Total Probability:
\[P(A) = P(A|B) \cdot P(B) + P(A|B') \cdot P(B')\]
\[P(A) = 0.40 \cdot 0.29 + 0 \cdot (1 - 0.29)\]
\[P(A) = 0.116 + 0\]
\[P(A) = 0.116\]
Therefore,
\[P(B|A) = \frac{0.116}{0.116}\]
\[P(B|A) = 1\]
Therefore, the probability that a person in one of these positions has a bachelor's degree is 100%, which is not one of the provided answer choices. This might be an error in the calculation or the data provided.