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!