Statistics Definitions > Conditional Probability

Conditional probability can be thought of as looking at the probability of one event occurring with some relationship to one or more other events. For example:

- Event A is that it is raining outside, and it has a 0.3 (30%) chance of raining today.
- Event B is that you will need to go outside, and that has a probability of 0.5 (50%).

A conditional probability would look at these two events in relationship with one another, such as the probability that it is both raining *and *you will need to go outside.

The formula for conditional probability is:

P(B|A) = P(A and B) / P(A)

which can also be rewritten as:

P(B|A) = P(A∩B) / P(A)

## Conditional Probability Formula Examples

Watch the video, or read the examples below:

**Example 1**. In a group of 100 sports car buyers, 40 bought alarm systems, 30 purchased bucket seats, and 20 purchased an alarm system and bucket seats. If a car buyer chosen at random bought an alarm system, what is the probability they also bought bucket seats?

Step 1: Figure out P(A). It’s given in the question as 40%, or 0.4.

Step 2: Figure out P(A∩B). This is the intersection of A and B: both happening together. It’s given in the question 20 out of 100 buyers, or 0.2.

Step 3: Insert your answers into the formula:

P(B|A) = P(A∩B) / P(A) = 0.2 / 0.4 = 0.5.

The probability that a buyer bought bucket seats, given that they purchased an alarm system, is 50%.

**Example 2: **This question uses the following contingency table:

What is the probability a randomly selected person is male, given that they own a pet?

Step 1: Repopulate the formula with new variables so that it makes sense for the question (optional, but it helps to clarify what you’re looking for). I’m going to say M is for male and PO stands for pet owner, so the formula becomes:

P(M|PO) = P(M∩PO) / P(PO)

Step 2: Figure out P(M∩PO) from the table. The intersection of male/pets (the intersection on the table of these two factors) is 0.41.

Step 3: Figure out P(PO) from the table. From the total column, 86% (0.86) of respondents had a pet.

Step 4: Insert your values into the formula:

P(M|PO) = P(M∩PO) / P(M) = 0.41 / 0.86 = 0.477, or 47.7%.

Why do we care about conditional probability? Events in life rarely have simple probability. Think about the probability of getting rainfall.

## Conditional Probability in Real Life

Conditional probability is used in many areas, including finance, insurance and politics. For example, the re-election of a president depends upon the voting preference of voters and perhaps the success of television advertising — even the probability of the opponent making gaffes during debates!

The weatherman might state that your area has a probability of rain of 40 percent. However, this fact is *conditional* on many things:

- The probability of a cold front coming to your area.
- The probability of rain clouds forming.
- The probability of another front pushing the rain clouds away.

We say that the **conditional probability** of rain occurring depends on all the above events.

## Where does the Conditional Probability Formula Come From?

The formula for conditional probability is derived from the probability multiplication rule, P(A and B) = P(A)*P(B|A). You may also see this rule as P(A∪B). The Union symbol (∪) means “and”, as in event A happening and event B happening.

Step by step, here’s how to derive the conditional probability equation from the multiplication rule:

**Step 1**: Write out the multiplication rule:

P(A and B) = P(A)*P(B|A)

**Step 2:** Divide both sides of the equation by P(A):

P(A and B) / P(A) = P(A)*P(B|A) / / P(A)

**Step 3**: Cancel P(A) on the right side of the equation:

P(A and B) / P(A) = P(B|A)

**Step 4**: Rewrite the equation:

P(B|A) = P(A and B) / P(A)

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