# P(A∩B) Formula

P(A∩B) formula is used to find the probability of both independent events “A” and "B" happening together. The symbol "∩" means intersection. The intersection of two or more two sets is the set of elements that are common to every set. When events are independent, we can use the multiplication rule, which states that the two events A and B are independent if the occurrence of one event does not change the probability of the other event. Let us understand P(A∩B) formula using solved examples.

## What Is P(A∩B) Formula?

P(A∩B) is the probability of both independent events “A” and "B" happening together. This formula is used to quickly predict the result. P(A∩B) formula can be written as P(A∩B) = P(A) × P(B). P(A∩B) formula is given as:

### P(A∩B) Formula

P(A∩B) = P(A) × P(B)

where,

- P(A∩B) = Probability of both independent events “A” and "B" happening together.
- P(A) = Probability of an event “A”
- P(B) = Probability of an event “B”

To calculate P(A∩B) for dependent events, we use the concept of conditional probability and rewrite the formula as,

P(A∩B) = P(A|B)P(B) or P(A∩B) = P(B|A)P(A)

**Break down tough concepts through simple visuals.**

## Examples Using P(A∩B) Formula

**Example 1: What is the probability of selecting a red card and a 6 when a card is randomly selected from a deck of 52 cards? Solve this by using the P(A∩B) formula.**

**Solution:**

To find: Probability of selecting a red card and a 6.

Let A and B be the individual 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 probability of getting a red card from a deck of 52 cards, P(A) = 26/52

Since the probability of getting a red card and the probability of getting a 6 are calculated individually here, therefore the total number of cards for both cases will be taken as 52.

Thus, probability of getting a 6 from a deck of 52 cards, P(B) = 4/52

Using the P(A∩B) formula,

P(A∩B) = P(A) × P(B)

P(A∩B) = 26/52 × 4/52

= (26×4)/(52×52)

= 2/52

= 1/26

**Answer: The required probability = 1 / 26.**

**Example 2: What is the probability of getting a 2 and 3 when a die is rolled? Solve this by using the P(A∩B) formula.**

**Solution:**

To find: The probability of getting a 2 and 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.

Using the P(A∩B) formula,

P(A∩B) = P(A) × P(B)

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.

**Answer:** **The required probability = 0.**

## FAQs on P(A∩B) Formula

### What Is P(A∩B) Formula?

P(A∩B) is the probability of both independent events “A” and "B" happening together, P(A∩B) formula can be written as P(A∩B) = P(A) × P(B),

where,

- P(A∩B) = Probability of both independent events “A” and "B" happening together.
- P(A) = Probability of an event “A”
- P(B) = Probability of an event “B”

### What Does (A ∩ B) Represent in P(A ∩ B) Formula?

(A ∩ B) in P(A ∩ B) Formula represents the intersection of two events A and B. Symbol '∩' denotes intersection.

### How To Apply P(A ∩ B) Formula?

We apply P(A ∩ B) formula to calculate the probability of two independent events A and B occurring together. It is given as, P(A∩B) = P(A) × P(B), where, P(A) is Probability of an event “A” and P(B) = Probability of an event “B”.

### How Do You Find the P(A ∩ B) Formula of Two Independent Events?

We can find the probability of the intersection of two independent events as, P(A∩B) = P(A) × P(B), where, P(A) is Probability of an event “A” and P(B) = Probability of an event “B” and P(A∩B) is Probability of both independent events “A” and "B" happening together.