A computer consulting firm presently has bids out on three projects. Let Ai = {awarded project i}, for i = 1, 2, 3, and suppose that P(A1) = 0.22, P(A2) = 0.26, P(A3) = 0.29, P(A1 ∩ A2) = 0.11, P(A1 ∩ A3) = 0.08, P(A2 ∩ A3) = 0.05, P(A1 ∩ A2 ∩ A3) = 0.01. Use the probabilities given above to compute the following probabilities, and explain in words the meaning of each one. (Round your answers to four decimal places.)
(a)    P(A2 | A1) = 
.5
 
Correct: Your answer is correct.


Explain this probability in words.
If the firm is awarded project 2, this is the chance they will also be awarded project 1.
This is the probability that the firm is awarded both project 1 and project 2.    
This is the probability that the firm is awarded either project 1 or project 2.
If the firm is awarded project 1, this is the chance they will also be awarded project 2.
Correct: Your answer is correct.

(b)    P(A2 ∩ A3 | A1) =

Let's work through part (b), which asks for \( P(A_2 \cap A_3 | A_1) \). This is the conditional probability that both projects 2 and 3 are awarded, given that project 1 has been awarded. We can calculate this using the definition of conditional probability:

\[
P(A_2 \cap A_3 | A_1) = \frac{P(A_1 \cap A_2 \cap A_3)}{P(A_1)}
\]

We are given the following information:
- \( P(A_1) = 0.22 \)
- \( P(A_1 \cap A_2 \cap A_3) = 0.01 \)

Now substitute these values into the formula:

\[
P(A_2 \cap A_3 | A_1) = \frac{0.01}{0.22} \approx 0.0455
\]

### Meaning in Words:
This probability represents the chance that the firm is awarded both project 2 and project 3, given that they have already been awarded project 1. In other words, if the firm wins project 1, there's about a 4.55% chance that they'll also win both projects 2 and 3.

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Do you want any further details on this, or have other questions?

Here are 5 questions related to this topic:
1. How would you compute \( P(A_1 \cup A_2) \) using the given information?
2. What is the interpretation of \( P(A_1 \cap A_3) \)?
3. How would you calculate \( P(A_2 \cup A_3 | A_1) \)?
4. What does the complement \( P(A_1^c) \) represent in this scenario?
5. How does the inclusion-exclusion principle help in finding the probability of the union of events?

**Tip:** Conditional probability focuses on how one event affects the likelihood of another. Always divide by the probability of the "given" event!

A computer consulting firm presently has bids out on three projects. Let Ai = {awarded project i}, for i = 1, 2, 3, and suppose that P(A1) = 0.22, P(A2) = 0.26, P(A3) = 0.29, P(A1 ∩ A2) = 0.11, P(A1 ∩ A3) = 0.08, P(A2 ∩ A3) = 0.05, P(A1 ∩ A2 ∩ A3) = 0.01. Use the probabilities given above to compute the following probabilities, and explain in words the meaning of each one. (Round your answers to four decimal places.)

Explore how to calculate conditional probabilities in project award scenarios. Learn how to compute P(A2 | A1) and P(A2 ∩ A3 | A1) using given probability values, applying core concepts in probability theory such as intersections and conditional probability. Suitable for high school and college-level students.

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