Conditional Probability Formula

The conditional probability P(A|B) is the probability of event A occurring given that event B has already occurred.

The Formula

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

When to use: If I know BB happened, what's the chance of AA? Updates probability with new info.

Quick Example

P(draw red∣already drew one red)P(\text{draw red} \mid \text{already drew one red}) changes because there's one fewer red.

Notation

P(A∣B)P(A|B) reads 'the probability of AA given BB'; the vertical bar means 'given that' — it names the event you already know happened. P(A∩B)P(A \cap B) is the chance both happen, and P(B)P(B) is the new, smaller denominator (once BB is given, only BB's outcomes are left to count). P(A∣B)P(A|B) and P(B∣A)P(B|A) usually differ because they divide by different totals: with 12 sport, 8 music, 5 both, P(sport∣music)=5/8P(\text{sport}|\text{music}) = 5/8 but P(music∣sport)=5/12P(\text{music}|\text{sport}) = 5/12.

What This Formula Means

The conditional probability P(A∣B)P(A|B) is the probability of event AA occurring given that event BB has already occurred.

If I know BB happened, what's the chance of AA? Updates probability with new info.

Formal View

P(A∣B)=P(A∩B)P(B)P(A|B) = \frac{P(A \cap B)}{P(B)} where P(B)>0P(B) > 0

Worked Examples

Example 1

medium
In a class of 3030 students, 1818 play soccer, 1212 play basketball, and 66 play both. If a student plays soccer, what is the probability they also play basketball?

Answer

P(B∣S)=13P(B \mid S) = \frac{1}{3}

First step

1
We need P(B∣S)=P(B∩S)P(S)P(B \mid S) = \frac{P(B \cap S)}{P(S)}.

See the full worked solution + why-it-works coaching

SetupKey insightWhy it worksCommon pitfallConnection

Unlock answer keys One Family plan — every worked solution, all subjects

Example 2

hard
A test for a disease is 95%95\% accurate (true positive rate) with a 3%3\% false positive rate. If 1%1\% of the population has the disease, what is the probability a person who tests positive actually has the disease?

Example 3

medium
Of 200200 surveyed, 120120 own a dog and 8080 own a cat; 5050 own both. P(cat∣dog)=?P(\text{cat}|\text{dog})=?

Common Mistakes

  • Dividing by the whole sample space — divide by P(B)P(B), the known condition, which shrinks the denominator.
  • Swapping P(A∣B)P(A|B) and P(B∣A)P(B|A) — read carefully which event is the 'given.'
  • Assuming independence to skip the formula — only set P(A∣B)=P(A)P(A|B)=P(A) if you have verified the events are independent.

Why This Formula Matters

Conditional probability is how reasoning updates with evidence — it powers medical test interpretation, Bayes' rule, and the formal definition of independence (P(A∣B)=P(A)P(A|B)=P(A)). Students who forget that the denominator becomes P(B)P(B) instead of 1 misread risk and overcount. Recognizing it by "Has some information already been revealed that shrinks the set of possible outcomes?" — rather than by familiar numbers — is what lets a student tell it apart from joint probability and independent events and reversed conditional p(b∣a)p(b|a) in a mixed problem set.

Frequently Asked Questions

What is the Conditional Probability formula?

The conditional probability P(A∣B)P(A|B) is the probability of event AA occurring given that event BB has already occurred.

How do you use the Conditional Probability formula?

If I know BB happened, what's the chance of AA? Updates probability with new info.

What do the symbols mean in the Conditional Probability formula?

P(A∣B)P(A|B) reads 'the probability of AA given BB'; the vertical bar means 'given that' — it names the event you already know happened. P(A∩B)P(A \cap B) is the chance both happen, and P(B)P(B) is the new, smaller denominator (once BB is given, only BB's outcomes are left to count). P(A∣B)P(A|B) and P(B∣A)P(B|A) usually differ because they divide by different totals: with 12 sport, 8 music, 5 both, P(sport∣music)=5/8P(\text{sport}|\text{music}) = 5/8 but P(music∣sport)=5/12P(\text{music}|\text{sport}) = 5/12.

Why is the Conditional Probability formula important in Math?

Conditional probability is how reasoning updates with evidence — it powers medical test interpretation, Bayes' rule, and the formal definition of independence (P(A∣B)=P(A)P(A|B)=P(A)). Students who forget that the denominator becomes P(B)P(B) instead of 1 misread risk and overcount. Recognizing it by "Has some information already been revealed that shrinks the set of possible outcomes?" — rather than by familiar numbers — is what lets a student tell it apart from joint probability and independent events and reversed conditional p(b∣a)p(b|a) in a mixed problem set.

What do students get wrong about Conditional Probability?

The procedure for conditional probability is the easy part; the trap is dividing by the whole sample space. Asking "Has some information already been revealed that shrinks the set of possible outcomes?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Conditional Probability formula?

Before studying the Conditional Probability formula, you should understand: probability, independent events.

← Back to Conditional Probability See All Examples Practice Problems