Conditional Probability — Definition, Formula & Examples

Conditional Probability

A probability that is computed based on the assumption that some event has already occurred. The probability of event B given that event A has occurred is written P(B|A).

Formula: $\displaystyle P(B|A) = \frac{P(A \text{ and } B)}{P(A)}$

Example: Suppose a bag contains two red balls and two green balls, all of which are identical except for color. One ball is drawn at random and set aside. Then a second ball is drawn at random. Given the events
$A$ = the first ball is red
$B$ = the second ball is red,
find $P(B|A)\text{.}$

Answer: $\displaystyle P(B|A) = \frac{1}{3}\text{.}$ Why? After a red ball is removed, there are two green balls left and only one red ball.

Alternatively, we could use the formula above. $P(B|A) = \frac{P(A \text{ and } B)}{P(A)} = \frac{P(\text{both are red})}{P(\text{first is red})} = \frac{1/6}{1/2} = \frac{1}{3}.$

Key Formula

P(B|A) = \frac{P(A \text{ and } B)}{P(A)}

Where:

$P(B|A)$ = The probability of event B occurring given that event A has occurred
$P(A \text{ and } B)$ = The probability that both events A and B occur
$P(A)$ = The probability of event A occurring (must be greater than 0)

Worked Example

Problem: A standard deck has 52 cards. You draw one card and learn it is a face card (Jack, Queen, or King). What is the probability that it is a King?

Identify the events: Let A = drawing a face card. Let B = drawing a King. Since every King is a face card, "A and B" simply means drawing a King.

Find the probabilities: There are 12 face cards and 4 Kings in a deck of 52 cards.

P(A) = \frac{12}{52}, \quad P(A \text{ and } B) = \frac{4}{52}

Apply the formula: Divide the probability of both events by the probability of the given event.

P(B|A) = \frac{P(A \text{ and } B)}{P(A)} = \frac{4/52}{12/52} = \frac{4}{12} = \frac{1}{3}

Answer: Given that the card is a face card, the probability it is a King is

\frac{1}{3}

Why It Matters

Conditional probability is essential whenever new information changes how likely an outcome is. It underpins medical testing (e.g., the probability you have a disease given a positive test), weather forecasting, and spam filters. It also forms the basis of Bayes' theorem, one of the most widely used results in statistics.

Common Mistakes

Mistake: Confusing P(B|A) with P(A|B).

Correction: These are generally not equal. The probability of rain given clouds is different from the probability of clouds given rain. Always check which event is the "given" condition.

Related Terms

Probability — The broader concept conditional probability builds on
Event — An outcome or set of outcomes in probability
Independent Events — Events where conditioning does not change the probability
Bayes' Theorem — Uses conditional probability to reverse the condition