# P(A or B) = P(A) + P(B) - P(A and B)

• Mar 26th 2011, 01:54 PM
Shogo39
P(A or B) = P(A) + P(B) - P(A and B)
I understand the equation P(A or B) = P(A) + P(B) - P(A and B)....I just don't know the logic behind it. Why do we subtract the P(A and B) will both happen if the event are independent of each other. Here is an example:

Mr. Pontofobe wants to drive from New Jersey to Long Island through Manhattan. He does not want to use the bridges though, so his only choices are the tunnels:

http://www.webassign.net/userimages/147097?db=v4net

Tunnels are closed for maintenance with the following probability:
Hudson Tunnel: 0.12
Lincoln Tunnel: 0.18
Brooklyn-Battery Tunnel: 0.08
Queens-Midtown Tunnel: 0.16

Assume that the tunnels close for maintenance independently.
Determine the probability that Mr. Pontofobe can't make his journey:

Why is it that we subtract the probability that all 4 tunnels will be closed if indeed he CANNOT make his trip if this happens? I'm not sure of the logic behind the equation.
• Mar 26th 2011, 02:06 PM
Plato
Quote:

Originally Posted by Shogo39
I understand the equation P(A or B) = P(A) + P(B) - P(A and B)....I just don't know the logic behind it. Why do we subtract the P(A and B) will both happen if the event are independent of each other.

It is simply the basic counting principle from set theory.
All probability is derived from set theory.

If we have two sets $A~\&~B$ if we want to know how elements are in $A\text{ or }B,~~A\cup B$ we count the number in $A$ and we count the number in $B$. If we add those together we get an over count because we have counted $A\cap B$ twice. So subtract.