The complete question is:
"A consulting firm is bidding on 3 projects. Let Ai= {Bid on project i is won}, i=1,2,3.
Suppose that: P(A1) = .22, P(A2) = .25, P(A3)=.28, P(A1 ∩ A2) = .11, P(A1 ∩ A3) = .05, P(A2 ∩ A3) = .07, P(A1 ∩ A2 ∩ A3) = .01
Find P(A^c1 ∩ A^c2 ∩ A3).
Could someone please explain how to find this, our professor said that we didn't need any formulas to calculate the probablities, but we've been looking at it for a while now and just can't figure this one problem out, and the next problem goes off of the answer to this one!
Thank you to anyone in advance!
"A consulting firm is bidding on 3 projects. Let Ai= {Bid on project i is won}, i=1,2,3.
Suppose that: P(A1) = .22, P(A2) = .25, P(A3)=.28, P(A1 ∩ A2) = .11, P(A1 ∩ A3) = .05, P(A2 ∩ A3) = .07, P(A1 ∩ A2 ∩ A3) = .01
Find P(A^c1 ∩ A^c2 ∩ A3).
Could someone please explain how to find this, our professor said that we didn't need any formulas to calculate the probablities, but we've been looking at it for a while now and just can't figure this one problem out, and the next problem goes off of the answer to this one!
Thank you to anyone in advance!
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Use the basic principle that for any events R and S,
P(R^c and S) + P(R and S) = P(S), which implies P(R^c and S) = P(S) - P(R and S).
Therefore, we have
P(A^c1 ∩ A^c2 ∩ A3)
= P(A^c1 ∩ A3) - P(A^c1 ∩ A2 ∩ A3)
= P(A3) - P(A1 ∩ A3) - [P(A2 ∩ A3) - P(A1 ∩ A2 ∩ A3)]
= 0.28 - 0.05 - (0.07 - 0.01)
= 0.17
P(R^c and S) + P(R and S) = P(S), which implies P(R^c and S) = P(S) - P(R and S).
Therefore, we have
P(A^c1 ∩ A^c2 ∩ A3)
= P(A^c1 ∩ A3) - P(A^c1 ∩ A2 ∩ A3)
= P(A3) - P(A1 ∩ A3) - [P(A2 ∩ A3) - P(A1 ∩ A2 ∩ A3)]
= 0.28 - 0.05 - (0.07 - 0.01)
= 0.17