Hi, I need help with the last question of a homework problem.

I am given: P(A1) = 0.23, P(A2) = 0.26, P(A3) = 0.29, P(A1 ∩ A2) = 0.08, P(A1 ∩ A3) = 0.07, P(A2 ∩ A3) = 0.05, P(A1 ∩ A2 ∩ A3) = 0.01.

I need to find: P(A1 ∩ A2 ∩ A3 | A1 ∪ A2 ∪ A3).

I know:

P(A1 ∪ A2 ∪ A3) =

P(A1) + P(A2) + P(A3) − P(A1 ∩ A2) − P(A1 ∩ A3) − P(A2 ∩ A3) + P(A1 ∩ A2 ∩ A3)

= .23 + .26 + .29 - .08 - .07 - .05 + .01 = .59

and, it is given:

P(A1 ∩ A2 ∩ A3) = 0.01, but I do not know how to combine these for P(A1 ∩ A2 ∩ A3 | A1 ∪ A2 ∪ A3). Can someone help please?

I know P(A | B) = P(A ∩ B) / P(B), but if I extrapolate that with the above, more complex formulas, it yields a completely incorrect answer.

I am given: P(A1) = 0.23, P(A2) = 0.26, P(A3) = 0.29, P(A1 ∩ A2) = 0.08, P(A1 ∩ A3) = 0.07, P(A2 ∩ A3) = 0.05, P(A1 ∩ A2 ∩ A3) = 0.01.

I need to find: P(A1 ∩ A2 ∩ A3 | A1 ∪ A2 ∪ A3).

I know:

P(A1 ∪ A2 ∪ A3) =

P(A1) + P(A2) + P(A3) − P(A1 ∩ A2) − P(A1 ∩ A3) − P(A2 ∩ A3) + P(A1 ∩ A2 ∩ A3)

= .23 + .26 + .29 - .08 - .07 - .05 + .01 = .59

and, it is given:

P(A1 ∩ A2 ∩ A3) = 0.01, but I do not know how to combine these for P(A1 ∩ A2 ∩ A3 | A1 ∪ A2 ∪ A3). Can someone help please?

I know P(A | B) = P(A ∩ B) / P(B), but if I extrapolate that with the above, more complex formulas, it yields a completely incorrect answer.

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