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:
Sample Problem
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
.
Sample Problem
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.
Therefore,
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)
as
1 = (probability A happens) + (probability "not A" happens).
Hold that formula in your brain for a sec. We'll make use of it, promise.
Sample Problem
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.
Example 1
You roll two dice. What is the probability of getting 2 on the first die or 3 on the second die? |
.
Example 2
Let A be the event of rolling 4 on a die. What's the probability of not rolling 4? |
Example 3
Bernard has a weighted coin. We thought he seemed shady. His coin lands heads up with probability |
. What is the probability his coin lands tails up?
Exercise 1
Determine if the two events are mutually exclusive.
Rolling even numbers on two dice and the dice summing to 7.
Exercise 2
Determine if the two events are mutually exclusive.
Pulling an ace from a deck of cards and, without replacing the ace, pulling another ace from the deck. Assuming the person pulling the aces is not David Copperfield.
Exercise 3
Determine if the two events are mutually exclusive.
Flipping two coins, having the first coin be heads and both coins be tails.
Exercise 4
For the following pair of events, (a) determine if the two events A and B are mutually exclusive, and (b) find the probability that A or B (or both) happens.
You flip one coin.
Event A: You get heads.
Event B: You get tails.
Exercise 5
For the following pair of events, (a) determine if the two events A and B are mutually exclusive, and (b) find the probability that A or B (or both) happens.
You roll two dice.
Event A: The first die shows an odd number.
Event B: The second die shows the number 6.
Exercise 6
For the following pair of events, (a) determine if the two events A and B are mutually exclusive, and (b) find the probability that A or B (or both) happens.
You roll two dice.
Event A: The dice sum to 5.
Event B: The dice sum to 8.
Hey, something's not adding up here...
Exercise 7
For the following pair of events, (a) determine if the two events A and B are mutually exclusive, and (b) find the probability that A or B (or both) happens.
Penny has a green shirt, a purple shirt, and a red shirt. She also has black pants and blue pants. She picks a shirt at random and picks a pair of pants at random. What an appearance-conscious fashionista she is.
Hey, it was dark when she got dressed. Cut her some slacks.
Event A: Penny wears a green shirt.
Event B: Penny wears black pants.
Exercise 8
For the following pair of events, (a) determine if the two events A and B are mutually exclusive, and (b) find the probability that A or B (or both) happens.
Same deal as above: Penny has a green shirt, a purple shirt, and a red shirt. She has black pants and blue pants. She picks a shirt at random and picks a pair of pants at random.
Event A: Penny wears a green shirt.
Event B: Penny wears a red shirt.
Exercise 9
A spinner for the game of Twister has four colors: red, blue, yellow, and green. If the spinner is a little wobbly and has a probability of 
of landing on red, what is the probability that the spinner doesn't land on red? Careful...your entire sense of balance depends on getting this question right.
Exercise 10
A frog will catch his next fly with probability 
. He could probably catch more flies with honey, but his pantry is currently empty. What is the probability that the frog will miss his next fly?
Exercise 11
Jenny has blue shoes and green shoes. She can only wear one color of shoes at a time, and she must wear shoes. She gave barefoot a try one day last week and it didn't work out so well. If the probability Jenny wears blue shoes is 
, what's the probability Jenny wears green shoes?
