How to calculate conditional probability

Answer

What is conditional probability?

Conditional probability is the probability of an event B occurring given the knowledge that an event A has already occurred. This probability is written as P(A|B), indicating the probability of event A when event B has already occurred.

The conditional probability formula

To calculate conditional probability, use the following formula:

$P(A|B)=\frac{P(A\cap B)}{P(B)}$

Where:

1. P(A|B) is the probability of event A occurring given that event B has occurred
2. P(A ∩ B) is the probability of both events A and B occurring simultaneously
3. P(B) is the probability of event B occurring

This formula is only valid when P(B) is greater than 0.

Step-by-step calculation process

To calculate conditional probability, follow these steps:

1. Identify event A (the event whose probability you want to find)
2. Identify event B (the condition or given event)
3. Calculate P(B), the probability of the given event
4. Calculate P(A ∩ B), the probability that both events occur together
5. Divide P(A ∩ B) by P(B) to get P(A|B)

Worked example

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: Identify the events.

• Event A: Buyer purchased bucket seats
• Event B: Buyer purchased an alarm system

Step 2: Calculate P(B).

$P(B)=\frac{40}{100}=0.4$

Step 3: Calculate P(A ∩ B).

$P(A\cap B)=\frac{20}{100}=0.2$

Step 4: Apply the conditional probability formula.

$P(A|B)=\frac{P(A\cap B)}{P(B)}=\frac{0.2}{0.4}=0.5$

The probability that a buyer purchased bucket seats, given that they bought an alarm system, is 0.5 or 50%.

Alternative calculation using counts

When working with equally likely outcomes, conditional probability can be calculated directly using counts:

$P(A|B)=\frac{n(A\cap B)}{n(B)}$

Where n(A ∩ B) is the number of outcomes in both A and B, and n(B) is the total number of outcomes in B.

Using the same example:

$P(A|B)=\frac{20}{40}=0.5$

This produces the same result of 50%.

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