Conditional Probability Formula

The Formula

P(A|B) = \frac{P(A \text{ 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(\text{draw red} \mid \text{already drew one red}) changes because there's one fewer red.

Notation

P(A|B) reads 'probability of A given B'; the vertical bar means 'given that'

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) = \frac{P(A \cap 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?

Solution

  1. 1
    We need P(B \mid S) = \frac{P(B \cap S)}{P(S)}.
  2. 2
    P(B \cap S) = \frac{6}{30} = \frac{1}{5} and P(S) = \frac{18}{30} = \frac{3}{5}.
  3. 3
    P(B \mid S) = \frac{1/5}{3/5} = \frac{1}{3}.

Answer

P(B \mid S) = \frac{1}{3}
Conditional probability restricts the sample space to only those outcomes where the given condition is true. Here we only consider the 18 soccer players.

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?

Common Mistakes

  • Swapping the condition: treating P(A|B) as if it were P(B|A)
  • Using the total sample size as the denominator instead of the size of the given condition subset
  • Forgetting that P(A|B) restricts the sample space to only outcomes where B occurred

Why This Formula Matters

Conditional probability is fundamental to Bayes' theorem, medical testing, and any reasoning where new information changes what you know about an outcome.

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 'probability of A given B'; the vertical bar means 'given that'

Why is the Conditional Probability formula important in Math?

Conditional probability is fundamental to Bayes' theorem, medical testing, and any reasoning where new information changes what you know about an outcome.

What do students get wrong about Conditional Probability?

P(A|B) \neq P(B|A). P(\text{disease}|\text{positive test}) \neq P(\text{positive test}|\text{disease}).

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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