Mutually Exclusive Events Examples

You roll two dice. What is the probability of getting 2 on the first die or 3 on the second die?

The two events here are:

A: rolling 2 on the first die.
B: rolling 3 on the second die.

These events are not mutually exclusive. We can roll 2 on the first die and 3 on the second die at the same time. We've done it before, and we shall live to see it happen again. We can find the probability that at least one of the events A or B happens by looking at the table:

If we roll 2 on the first die, 3 on the second die, or both (2 on the first and 3 on the second at the same time), then (at least) one of the events A and B has happened. "At least one" is good enough for us. We're easy. There are 11 favorable outcomes where A or B happens, out of 36 total outcomes. So the probability of rolling 2 on the first die or 3 on the second die (or both) is .

If instead we try to add the probabilities of the two events, we find

which is wrong. Close, but wrong. Like MapQuest directions.

Let A be the event of rolling 4 on a die. What's the probability of not rolling 4?

The probability that we don't roll 4 is equal to:

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?

If event A is landing heads up, then "not A" is landing tails up.

To summarize, if you know the probability that A happens and you want to know the probability A doesn't, take

1 – (probability of A).