To find the probability that Jimmy meets Kelvin's expectations based on the information given, we can use Bayes' theorem.

Step 1: Define the Events

Let:

• $S$: Jimmy meets Kelvin's expectations.
• $F$: Jimmy does not meet Kelvin's expectations.
• $C$: Jimmy completed a project before the deadline.

We need to find $P(S|C)$, the probability that Jimmy meets Kelvin's expectations given that he completed a project before the deadline.

Step 2: Identify Probabilities

From the problem:

• Probability of meeting expectations: $P(S) = \frac{6}{8} = 0.75$.
• Probability of not meeting expectations: $P(F) = \frac{2}{8} = 0.25$.
• Probability of completing the project on time if meeting expectations: $P(C|S) = 0.85$.
• Probability of completing the project on time if not meeting expectations: $P(C|F) = 0.35$.

Step 3: Use Bayes' Theorem

Bayes' theorem states:

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

where $P(C)$ is the total probability of completing the project on time, given by:

$P(C) = P(C|S) \times P(S) + P(C|F) \times P(F)$

Substituting the values:

$P(C) = (0.85 \times 0.75) + (0.35 \times 0.25) = 0.6375 + 0.0875 = 0.725$

Now, substitute $P(C)$ into Bayes' theorem:

$P(S|C) = \frac{0.85 \times 0.75}{0.725} = \frac{0.6375}{0.725} \approx 0.8793$

Conclusion

The probability that Jimmy’s performance meets Kelvin's expectations, given that he successfully completed a project before the deadline, is approximately 0.8793 or 87.93%.

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Would you like more details or have any questions about the calculations? Here are some related questions:

1. How would the probability change if Jimmy completed two projects on time?
2. What if the probability of on-time completion for unsatisfactory engineers was 50%?
3. How does changing the number of satisfactory engineers affect this probability?
4. What are the implications of hiring decisions based on probability?
5. How would this model adjust if engineers' performances were re-evaluated?

Tip: Bayes' theorem is particularly useful when updating the probability of an event based on new evidence.