Conditional Probability Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Conditional Probability.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Given B, you're only considering the subset where B occurred.

Common stuck point: P(A|B) \neq P(B|A). P(\text{disease}|\text{positive test}) \neq P(\text{positive test}|\text{disease}).

Sense of Study hint: Shrink your sample space to only the cases where the 'given' event happened. Now count the favorable cases within that smaller group.

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?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
Two cards are drawn without replacement from a standard deck. Given that the first card is a king, what is the probability the second card is also a king?

Example 2

hard
A jar contains 6 red marbles and 4 blue marbles. Two marbles are drawn without replacement. Given that at least one marble is blue, what is the probability both marbles are blue?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

probabilityindependent events