The probability of two events that are not mutually exclusive may be calculated using the formula

Given two events, A and B, to “find the probability of A or B” means to find the probability that either event A or event B occurs.

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Examples: P(A∪B) for Mutually Exclusive Events
Examples: P(A∪B) for Not Mutually Exclusive Events
1. What is the Probability of A and B?
2. What is the Probability of A or B?

We typically write this probability in one of two ways:

P(A or B) – Written form
P(A∪B) – Notation form

The way we calculate this probability depends on whether or not events A and B are mutually exclusive or not. Two events are mutually exclusive if they cannot occur at the same time.

If A and B are mutually exclusive, then the formula we use to calculate P(A∪B) is:

Mutually Exclusive Events: P(A∪B) = P(A) + P(B)

If A and B are not mutually exclusive, then the formula we use to calculate P(A∪B) is:

Not Mutually Exclusive Events: P(A∪B) = P(A) + P(B) - P(A∩B)

Note that P(A∩B) is the probability that event A and event B both occur.

The following examples show how to use these formulas in practice.

Examples: P(A∪B) for Mutually Exclusive Events

Example 1: What is the probability of rolling a dice and getting either a 2 or a 5?

Solution: If we define event A as getting a 2 and event B as getting a 5, then these two events are mutually exclusive because we can’t roll a 2 and a 5 at the same time. Thus, the probability that we roll either a 2 or a 5 is calculated as:

P(A∪B) = (1/6) + (1/6) = 2/6 = 1/3.

Example 2: Suppose an urn contains 3 red balls, 2 green balls, and 5 yellow balls. If we randomly select one ball, what is the probability of selecting either a red or green ball?

Solution: If we define event A as selecting a red ball and event B as selecting a green ball, then these two events are mutually exclusive because we can’t select a ball that is both red and green. Thus, the probability that we select either a red or green ball is calculated as:

P(A∪B) = (3/10) + (2/10) = 5/10 = 1/2.

Examples: P(A∪B) for Not Mutually Exclusive Events

The following examples show how to calculate P(A∪B) when A and B are not mutually exclusive events.

Example 1: If we randomly select a card from a standard 52-card deck, what is the probability of choosing either a Spade or a Queen?

Solution: In this example, it’s possible to choose a card that is both a Spade and a Queen, thus these two events are not mutually exclusive.

If we let event A be the event of choosing a Spade and event B be the event of choosing a Queen, then we have the following probabilities:

P(A) = 13/52
P(B) = 4/52
P(A∩B) = 1/52

Thus, the probability of choosing either a Spade or a Queen is calculated as:

P(A∪B) = P(A) + P(B) – P(A∩B) = (13/52) + (4/52) – (1/52) = 16/52 = 4/13.

Example 2: If we roll a dice, what is the probability that it lands on a number greater than 3 or an even number?

Solution: In this example, it’s possible for the dice to land on a number that is both greater than 3 and even, thus these two events are not mutually exclusive.

If we let event A be the event of rolling a number greater than 3 and event B be the event of rolling an even number, then we have the following probabilities:

P(A) = 3/6
P(B) = 3/6
P(A∩B) = 2/6

Thus, the probability that the dice lands on a number greater than 3 or an even number is calculated as:

P(A∪B) = P(A) + P(B) – P(A∩B) = (3/6) + (3/6) – (2/6) = 4/6 = 2/3.

Watch the video for a few quick examples of how to find the Probability of A and B / A or B:

Probability of A or B (also A and B)

Watch this video on YouTube.

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You may want to read this article first: Dependent or Independent Event? How to Tell the Difference.

Probability of A and B.
Probability of A or B.

A Venn diagram intersection shows events a and b happening together.

1. What is the Probability of A and B?

The probability of A and B means that we want to know the probability of two events happening at the same time. There’s a couple of different formulas, depending on if you have dependent events or independent events.

Formula for the probability of A and B (independent events): p(A and B) = p(A) * p(B).

If the probability of one event doesn’t affect the other, you have an independent event. All you do is multiply the probability of one by the probability of another.

Examples

Example 1: The odds of you getting promoted this year are 1/4. The odds of you being audited by the IRS are about 1 in 118. What are the odds that you get promoted and you get audited by the IRS?

Solution:
Step 1: Multiply the two probabilities together: p(A and B) = p(A) * p(B) = 1/4 * 1/118 = 0.002.

That’s it!

Example 2: The odds of it raining today is 40%; the odds of you getting a hole in one in golf are 0.08%. What are your odds of it raining and you getting a hole in one?

Solution:
Step 1: Multiply the probability of A by the probability of B. p(A and B) = p(A) * p(B) = 0.4 * 0.0008 = 0.00032.

That’s it!

Formula for the probability of A and B (dependent events): p(A and B) = p(A) * p(B|A)

The formula is a little more complicated if your events are dependent, that is if the probability of one event effects another. In order to figure these probabilities out, you must find p(B|A), which is the conditional probability for the event.

Example question: You have 52 candidates for a committee. Four are persons aged 18 to 21. If you randomly select one person, and then (without replacing the first person’s name), randomly select a second person, what is the probability both people will be between 18 and 21 years old?

Solution:
Step 1: Figure out the probability of choosing an 18 to 21 year old on the first draw. As there are 52 possibilities, and 4 are aged 18 to 21, you have a 4/52 = 1/13 chance.

Step 2: Figure out p(B|A), which is the probability of the next event (choosing a second person aged 18 to 21) given that the first event in Step 1 has already happened.
There are 51 people left, and only 3 are aged 18 to 21 now, so the probability of choosing a young adult again is 3/51 = 1 / 17.

Step 3: Multiply your probabilities from Step 1(p(A)) and Step 2(p(B|A)) together:
p(A) * p(B|A) = 1/13 * 1/17 = 1/221.

Your odds of choosing two people aged 18 to 21 are 1 out of 221.

2. What is the Probability of A or B?

The probability of A or B depends on if you have mutually exclusive events (ones that cannot happen at the same time) or not.

If two events A and B are mutually exclusive, the events are called disjoint events. The probability of two disjoint events A or B happening is:

p(A or B) = p(A) + p(B).

Example question: What is the probability of choosing one card from a standard deck and getting either a Queen of Hearts or Ace of Hearts? Since you can’t get both cards with one draw, add the probabilities:
P(Queen of Hearts or Ace of Hearts) = p(Queen of Hearts) + p(Ace of Hearts) = 1/52 + 1/52 = 2/52.

If the events A and B are not mutually exclusive, the probability is:

(A or B) = p(A) + p(B) – p(A and B).

Example question: What is the probability that a card chosen from a standard deck will be a Jack or a heart?
Solution:

p(Jack) = 4/52
p(Heart) = 13/52
p(Jack of Hearts) = 1/52

So:
p(Jack or Heart) = p(Jack) + p(Heart) – p(Jack of Hearts) = 4/52 + 13/52 – 1/52 = 16/52.

References

Salkind, N. (2019). Statistics for People Who (Think They) Hate Statistics 7th Edition. SAGE.

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