Mutually Exclusive Events
These aren't parties that only mutual fund managers are allowed to attend. Don't worry; you aren't missing much. Not too much roof-raising goes on at those shindigs.
Sometimes two events can't happen at the same time. For example, we can't roll one die and get both an odd number and an even number on the same roll. We also can't roll two dice and get 1 on the first die, but a sum of 8 from the two dice. When events disagree like that, we call them mutually exclusive. We try not to call them that right to their faces, however, as we've already mentioned they're sort of disagreeable.
When two events A and B are mutually exclusive, to find the probability that either A or B happens, we add the probabilities of A and B. While this may seem like a new trick, we must confess we're being sneaky; you've actually done this already when finding probabilities like the one in the next example. We won't draw out the suspense any longer...here she is:
What's the probability of rolling a 5 or a 6 on a die?
We can't roll 5 and 6 at the same time, so these events are mutually exclusive. The probability of rolling 5 OR 6 on a die is then
The probability that Jenny wears purple shoes is . The probability that Jenny wears green shoes is . What is she, prepping for Mardi Gras? If Jenny can only wear one color of shoes, what is the probability that she wears either green or purple shoes?
She can't wear green and purple at once, no matter how many feet she has. The event of wearing purple shoes and the event of wearing green shoes are mutually exclusive. Good thing, because we feel like we saw this exact same issue addressed once on What Not to Wear.
To find the probability that either one happens, we add the individual probabilities:
When we talk about finding the probability that "A or B" happens, what we actually mean is the probability that A happens, B happens, or both A and B happen. When the events A and B are mutually exclusive it's impossible for both A and B to happen, so we happen upon this nice shortcut:
(probability A or B happens) = (probability of A) + (probability of B)
When events A and B are not mutually exclusive, both A and B could happen. Fingers crossed. This means (for now), when A and B are not mutually exclusive, we're going to go back to the formula
Here's a useful tidbit about probability: for an event A,
(probability A happens) + (probability A doesn't happen) = 1.
Event A and Event "not A" are mutually exclusive: we can't have A happen and not happen at the same time. The only time such a paradox was created that we can think of was when Keanu Reeves became an actor.
(probability A happens or "not A" happens) = (probability A happens) + (probability "not A" happens).
On the other hand, look at the left-hand side of that equation:
(probability A happens or "not A" happens).
What are the favorable outcomes for this event? No matter what, either A happens, or it doesn't. Every outcome is favorable. We have what we like to call a "win-win." Eat your heart out, Charlie Sheen.
Since (probability A happens or "not A" happens) always equals 1, we can rewrite the equation
(probability A happens or "not A" happens) = (probability A happens) + (probability "not A" happens)
1 = (probability A happens) + (probability "not A" happens).
Hold that formula in your brain for a sec. We'll make use of it, promise.
Let A be the event of rolling an odd number on a die. Then "not A" is the event of rolling an even number.
It's useful to know
1 = (probability A happens) + (probability "not A" happens)
(told ya!), because if we rearrange the equation by subtracting (probability A happens) from each side, we get
1 – (probability A happens) = (probability "not A" happens).
If we know the probability that an event happens, we also know the probability that the event doesn't happen: just subtract it from 1. Again, this discovery may seem intuitive to you, but it's always nice to back up intuition with some good old-fashioned formulas.