Question

Probability of exactly one of three independent events

Answer 1

As title states, I have 3 events X, Y, Z in a sample space each with a different probability of occurring, and I wanted to find the probability of exactly one of them happening. For reference, Z is independent of X and Y, while X and Y are dependent.

I know for just two events (say X and Y), I would use the formula: P(X)+P(Y)−2P(XY)P(X)+P(Y)−2P(XY) Is there a similar way to solve for finding one of three events? Thanks.

Answer 2

To simplify notation.
P(ABcCc)=P(ABc)−P(ABcC)P(ABcCc)=P(ABc)−P(ABcC)
=P(A)−P(AB)−[P(AC)−P(ABC)]=P(A)−P(AB)−[P(AC)−P(ABC)]
=P(A)−P(AB)−P(AC)+P(ABC)=P(A)−P(AB)−P(AC)+P(ABC)

Do that twice more for P(AcBCc) & P(AcBcC)P(AcBCc) & P(AcBcC) and add.