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)

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

Quick Example

P(draw red∣already drew one red) changes because there's one fewer red.

Notation

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

What This Formula Means

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

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

Formal View

P(A∣B)=P(A∩B)P(B) where P(B)>0

Worked Examples

Example 1

medium
In a class of 30 students, 18 play soccer, 12 play basketball, and 6 play both. If a student plays soccer, what is the probability they also play basketball?

Answer

P(B∣S)=13

First step

1
We need P(B∣S)=P(B∩S)P(S).

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Example 2

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

Example 3

medium
Of 200 surveyed, 120 own a dog and 80 own a cat; 50 own both. P(cat∣dog)=?

Common Mistakes

  • Dividing by the whole sample space — divide by P(B), the known condition, which shrinks the denominator.
  • Swapping P(A∣B) and P(B∣A) — read carefully which event is the 'given.'
  • Assuming independence to skip the formula — only set 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)). Students who forget that the denominator becomes 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) in a mixed problem set.

Frequently Asked Questions

What is the Conditional Probability formula?

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

How do you use the Conditional Probability formula?

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

What do the symbols mean in the Conditional Probability formula?

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

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