The procedure of getting the probability of the union of two events A B is P A P B P A ∩ B

Table of Contents Show

1. What is the Probability of A and B?
2. What is the Probability of A or B?
What is P(A∪B) Formula?
Examples Using P(A∪B) Formula
What is '∪' in P(A∪B) Formula?
What Is P(A∪B) Formula For Independent Events?

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)

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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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The symbol "∪" (union) means "or". i.e., P(A∪B) is the probability of happening of the event A or B. To find, P(A∪B), we have to count the sample points that are present in both A and B. So is P(A∪B) = P(A) + P(B)? No, because while counting the sample points from A and B, the sample points that are in A∩B are counted twice. Thus, we need to subtract P(A∩B) from the above sum to get P(A∪B).

P(A∪B) = P(A) + P(B) - P(A∩B)

What is P(A∪B) Formula?

From the above explanation, the P(A∪B) formula is:

P(A∪B) = P(A) + P(B) - P(A∩B)

This is also known as the addition theorem of probability.

But what if events A and B are mutually exclusive? In that case, P(A∩B) = 0.

The P(A∪B) formula when A and B are mutually exclusive is,

P(A∪B) = P(A) + P(B)

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Examples Using P(A∪B) Formula

Example 1: What is the probability of selecting a red card or a 6 when a card is randomly selected from a deck of 52 cards?

Solution:

To find: The probability of selecting a red card or a 6.

Let A and B be the probabilities of getting a red card and getting a 6 respectively.

We know that the number of red cards = 26,

The number of 6 labeled cards = 4, and

The number of red cards that are labeled 6 = 2.

Therefore,

The probability of getting a red card, P(A) = \(\dfrac{26}{52}\)

The probability of getting a 6, P(B) = \(\dfrac{4}{52}\)

The probability of getting both a Red and a 6, P(A∩B) = \(\dfrac{2}{52}\).

Using the P(A∪B) formula,

P(A∪B) = P(A) + P(B) - P(A∩B)

\(P(A∪B)= \dfrac{26}{52}+\dfrac{4}{52}-\dfrac{2}{52}\\[0.2cm]= \dfrac{30-2}{52}\\[0.2cm]= \dfrac{28}{52}\\[0.2cm]= \dfrac{7}{13}\)

Answer: The required probability = 7 / 13.

Example 2: What is the probability of getting a 2 or 3 when a die is rolled?

Solution:

To find: The probability of getting a 2 or 3 when a die is rolled.

Let A and B be the events of getting a 2 and getting a 3 when a die is rolled.

Then, P(A) = 1 / 6 and P(B) = 1 / 6.

In this case, A and B are mutually exclusive as we cannot get 2 and 3 in the same roll of a die.

Hence, P(A∩B) = 0.

Using the P(A∪B) formula,

P(A∪B) = P(A) + P(B) - P(A∩B)

P(A∪B) = 1 / 6 + 1 / 6 - 0 = 2 / 6 = 1 / 3

Answer: The required probability = 1 / 3.

he P(A∪B) formula is given as, P(A∪B) = P(A) + P(B) - P(A∩B), where P(A) is Probability of event A happening, P(B) is Probability of event B happening, and P(A∩B) is Probability of happening of both A and B.

What is '∪' in P(A∪B) Formula?

'∪' in P(A∪B) Formula represents the union of events A and event B.

What Is P(A∪B) Formula For Independent Events?

The P(A∪B) Formula for independent events is given as, P(A∪B) = P(A) + P(B), where P(A) is Probability of event A happening and P(B) is Probability of event B happening.