Question

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) = 0.25, P(A3) = 0.28, P(A1 ∩ A2) = 0.07, P(A1 ∩ A3) = 0.08, P(A2 ∩ A3) = 0.09, 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) =_________
Explain this probability in words.

1. If the firm is awarded project 1, this is the chance they will also be awarded project 2.
2. If the firm is awarded project 2, this is the chance they will also be awarded project 1.
3. This is the probability that the firm is awarded either project 1 or project 2.
4. This is the probability that the firm is awarded both project 1 and project 2.

b. P(A2 ∩ A3 | A1) =

Explain this probability in words.

1. If the firm is awarded projects 2 and 3, this is the chance they will also be awarded project 1.
2. This is the probability that the firm is awarded at least one of the projects.
3. If the firm is awarded project 1, this is the chance they will also be awarded projects 2 and 3.
4. This is the probability that the firm is awarded projects 1, 2, and 3.

c. P(A2 ∪ A3 | A1) =

Explain this probability in words.
1. This is the probability that the firm is awarded at least one of the projects.
2. This is the probability that the firm is awarded projects 1, 2, and 3.
3. If the firm is awarded at least one of projects 2 and 3, this is the chance they will also be awarded project 1.
4. If the firm is awarded project 1, this is the chance they will also be awarded at least one of the other two projects.


d. P(A1 ∩ A2 ∩ A3 | A1 ∪ A2 ∪ A3) =

Explain this probability in words.
1. If the firm is awarded at least two of the projects, this is the chance that they will be awarded all three projects.
2. This is the probability that the firm is awarded at least one of the projects.
3. If the firm is awarded at least one of the projects, this is the chance that they will be awarded all three projects.
4. This is the probability that the firm is awarded projects 1, 2, and 3.

Answer 1

a.

1.

b.

3.

c.

4.

d.

3.

Explanation:

a.

1. Is correct.

Remember that when we talk about
`P(X|Y)` that is the probability of event X given Y.

b.

3. is correct.

Following the same idea about conditional probabilities from the previous problem it is also important to understand that when you have the intersection of two events, the two are occurring.

c.

4.

Following the same idea about conditional probabilities from the first problem it is also important to understand that when you have the union of two events, at least one of them is occurring (both could happen).

d.

3.

Merging all ideas so far, the first union implies that at least one of the events are occurring and the intersection of all events means that all of them are occurring.