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# 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 Mutually Exclusive Outcomes Adding Probabilities - Not Mutually Exclusive    Like us on Facebook to get updates about new resources