Games
Problems
Go Pro!

# Adding Probabilities - Mutually Exclusive Outcomes

Reference > Mathematics > Probability

If we want to know the probability of two mutually exclusive outcomes happening, we have a simple formula:

P(A or B) = P(A) + P(B)

We read this as "The probability of either A or B happening is equal to the probability of A happening, plus the probability of B happening."

Let's try an example:

Example One
When I roll the six-sided die, what is the probability that I will either get a one or a four?

Solution
These are mutually exclusive outcomes, so we can use the formula shown above:

Outcome A is getting a one, and outcome B is getting a four.

P(A or B) = P(A) + P(B) =
1
6
+
1
6
=
2
6
=
1
3
.

Notice that this is the outcome we would have obtained using our P =
desired outcomes
possible outcomes
formula. So this is simply a new way of doing a problem you already knew how to do!

Example Two
I draw a card from the deck of cards. What is the probability that it is either a king or a heart?

Solution
These are not mutually exclusive, since a card could be both a king and a heart. Thus, we can't solve it using this formula. (We could solve it using the
desired
possible
formula, but for now we'll just move on.)

Example Three
I randomly pick a letter from the alphabet. What is the probability that it is either a J, a K, or a vowel.

Solution
These are mutually exclusive events, since none of the vowels are either J or K. We'll use an extended version of our formula:

P(J or K or Vowel) = P(J) + P(K) + P(Vowel) =
1
26
+
1
26
+
5
26
=
7
26

Example Four
I randomly pick a number between 50 and 200 inclusive. What is the probability that the number is either a two-digit odd number, or a three digit even number?

Solution
These are mutually exclusive events, since no number is both a two-digit number and a three-digit number.

P(O + E) = P(O) + P(E)

There are 151 possibilities, since we're including both 50 and 200.

The probability of getting a two-digit odd number is
25
151

The probability of getting a three-digit even number is
51
151
.

P(O + E) =
25
151
+
51
151
=
76
151

For each scenario below, either solve it using the formula given here (if the outcomes are mutually exclusive), or state that the outcomes are not mutually exclusive.

## Questions

1.
If I roll a die, what is the probability that the result will be either an odd number or 6?
2.
If I roll a die, what is the probability that the result will be either an odd number or a prime number?
3.
If I pick a letter from the alphabet, what is the probability that the letter will be either a consonant or an A?
4.
If I pick a card from the deck, what is the probability that it is either black or a red five?
5.
If I pick a card from the deck, what is the probability that it is either black or a five?
6.
If I draw a number from a hat containing all the two digit numbers, what is the probability that the number will either have repeated digits or end in zero?
7.
If I draw a number from a hat containing all the two digit numbers, what is the probability that the number will either have repeated digits, or will end in five?
Assign this reference page