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) = 0.23, P(A2) = 0.25, P(A3) = 0.29, P(A1 ∩ A2) = 0.09, P(A1 ∩ A3) = 0.11, P(A2 ∩ A3) = 0.07, P(A1 ∩ A2 ∩ A3) = 0.02. 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)
b. P(A2 ∩ A3 | A1)
c. P(A2 ∪ A3 | A1)
d. P(A1 ∩ A2 ∩ A3 | A1 ∪ A2 ∪ A3)

Answer 1

Step-by-step explanation:

The data below is what was provided in the question and it is what I solved the question with

P(A1) = 0.23

P(A2) = 0.25

P(A3) = 0.29

P(A1 n A2 ) = 0.09

P(A1 n A3) = 0.11

P(A2 n A3) = 0.07

P(A1 n A2 n A3) = 0.02

a

P(A2|A1) = P(A1 n A2)/P(A1)

= 0.09/0.23

= 0.3913

We have 39.13% confidence that event A2 will occur given that event A1 already occured

b.)

P(A3 n A3|A1) = P(A2 n A3)n A1)/P(A1)

= 0.02/0.23

= 0.08695

We have about 8.7% chance of events A2 and A3 occuring given that A1 already occured.

C.

P(A2 u A3|A1)

= P(A1 n A2)u(A1 n A3)/P(A1)

= P( A1 n A2) + P(A1 n A3) - P(A1 n A2 n A3) / P(A1)

= (0.09+0.11-0.02)/0.23

= 0.18/0.23

= 0.7826

We have 78.26% chance of A2 or A3 happening given that A1 has already occured.