To solve this, we use Bayes' Theorem. Let’s define the events:

• $A$: The person has at least a B.S. degree.
• $B$: The person earns more than $100,000.

We are tasked to find $P(A|B)$, the probability that a person has at least a B.S. degree given that they earn more than $100,000.

Bayes' Theorem Formula:

$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$

Step 1: Determine $P(A)$, $P(B|A)$, and $P(B)$

• $P(A) = 0.2$ (Probability of having at least a B.S. degree).
• $P(B|A) = 0.6$ (Probability of earning more than $100,000 given at least a B.S. degree).
• $P(B|A^c) = 0.2$ (Probability of earning more than $100,000 without a B.S. degree).

Here, $A^c$ is the complement of $A$, meaning the person does not have a B.S. degree. The probability of not having a B.S. degree is: $P(A^c) = 1 - P(A) = 0.8$

Step 2: Calculate $P(B)$ (Total probability of earning more than $100,000)

Using the law of total probability, $P(B)$ is given by: $P(B) = P(B|A) \cdot P(A) + P(B|A^c) \cdot P(A^c)$ Substituting values: $P(B) = (0.6 \cdot 0.2) + (0.2 \cdot 0.8) = 0.12 + 0.16 = 0.28$

Step 3: Apply Bayes' Theorem

Now, substitute into Bayes' formula: $P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$ $P(A|B) = \frac{(0.6 \cdot 0.2)}{0.28} = \frac{0.12}{0.28} \approx 0.4286$

Final Answer:

The probability that a person has at least a B.S. degree given that they earn more than $100,000 is approximately: $\boxed{0.4286}$

Would you like a detailed explanation or further breakdown? Here are 5 related questions to consider:

1. How does Bayes' Theorem apply in other real-world situations involving conditional probabilities?
2. What if the probabilities of earning more than $100,000 were switched—how would that affect the result?
3. How does the law of total probability help in breaking down complex problems?
4. What is the significance of complementary probabilities like $P(A^c)$ in such problems?
5. How would the result change if $P(A)$ increased to 0.5?

Tip: Always ensure that probabilities in such problems sum up correctly across all possible events to avoid logical inconsistencies.