General Addition Rule for Probability

[Jason76]

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.

 

[srmichael]

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.