If \(\displaystyle A\) and \(\displaystyle B\) are any two events then the \(\displaystyle P(A\) or \(\displaystyle B) = P(A) + P(B) - P(A + B)\)What does this mean? How does this relate to the bell curve when finding probability?
\(\displaystyle P(A) + P(B) - P(A + B)\)
Wouldn't this mean \(\displaystyle 0\)?
similar to \(\displaystyle 2(4) + 2(5) - 2(4 + 5) = 0\) in it's form.
Wait, what????
P is not a variable in the sense that P(A) does not mean P times A. P(A) means the probability that event A occurs. And the last term is not P(A + B) it is supposed to be P(A and B) or also written as \(\displaystyle P(A \cap B)\). Similarly, P(A or B) is also written as \(\displaystyle P(A \cup B)\)
So the formula is: \(\displaystyle P(A \cup B) = P(A) + P(B) - P(A \cap B)\) which means that if you have two events, A and B, that are not mutually exclusive, then the probablity that A or B occurs is the probability that A occurs plus the probability that B occurs minus the probability that A and B occurs.
Think of it in terms of a Venn diagram. When you add P(A) + P(B) you have added the overlapping part of the Venn diagram, which is the probability that both A and B occurs, twice. So you then have to subtract one of those probabilities out.
Example:
A = Pick a blue shirt out to wear
B = Pick a short sleeve shirt out to wear
A and B = pick out a blue short sleeve shirt
Let P(A) = 0.6
Let P(B) = 0.5
Let P(A and B) = 0.3
So the probability that you will pick out either a blue shirt or a short sleeve shirt is:
\(\displaystyle P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
\(\displaystyle P(A \cup B) = 0.6 + 0.5 - 0.3\)
\(\displaystyle P(A \cup B) = 0.8\)
So there is an 80% chance you will choose either a blue shirt or a short sleeve shirt to wear today.
