Bayes' Theorem with Tree Diagrams Cheat Sheet

A printable reference covering Bayes' theorem, conditional probability, tree diagrams, joint probability, and reverse conditional probabilities for grades 10-12.

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Bayes' theorem helps students update probabilities when new information is known. This cheat sheet covers how to use tree diagrams to organize conditional probabilities and reverse a probability statement correctly. It is especially useful for questions involving tests, surveys, risk, or classification. Students need it because many errors happen when they confuse $P(A\mid B)$ with $P(B\mid A)$ . The main idea is to multiply along branches of a tree diagram to find joint probabilities, then add relevant joint probabilities to find totals. Bayes' theorem uses the formula $P(A\mid B)=\frac{P(A)P(B\mid A)}{P(B)}$ to find the probability of a cause after seeing an outcome. The denominator $P(B)$ often comes from adding all paths that lead to event $B$ . Tree diagrams make each path visible, which helps students choose the correct numerator and denominator.

Key Facts

Conditional probability is written as $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$ , where $P(B)\neq 0$ .
Bayes' theorem is $P(A\mid B)=\frac{P(A)P(B\mid A)}{P(B)}$ .
The total probability formula is $P(B)=P(A)P(B\mid A)+P(A^c)P(B\mid A^c)$ for two complementary cases.
In a tree diagram, multiply probabilities along one complete path to find a joint probability such as $P(A\cap B)$ .
In a tree diagram, add the probabilities of all paths that lead to the same final event to find a total probability.
Complementary probabilities satisfy $P(A^c)=1-P(A)$ .
The numerator in Bayes' theorem is the joint probability of the condition and the event you want, such as $P(A\cap B)$ .
The denominator in Bayes' theorem is the total probability of the given condition, such as $P(B)$ .

Vocabulary

Bayes' theorem: A formula for finding a reverse conditional probability using $P(A\mid B)=\frac{P(A)P(B\mid A)}{P(B)}$ .
Conditional probability: The probability that event $A$ happens given that event $B$ has already happened, written $P(A\mid B)$ .
Tree diagram: A branching diagram that shows stages of an experiment and labels each branch with its probability.
Joint probability: The probability that two events both happen, written $P(A\cap B)$ .
Complement: The event that $A$ does not happen, written $A^c$ , with probability $P(A^c)=1-P(A)$ .
Total probability: The probability of an outcome found by adding the probabilities of all possible paths that produce that outcome.

Common Mistakes to Avoid

Confusing $P(A\mid B)$ with $P(B\mid A)$ is wrong because the condition changes the sample space and usually gives a different probability.
Using only one tree path for the denominator is wrong because $P(B)$ must include every path that leads to event $B$ .
Adding probabilities along a path is wrong because a complete path represents events happening together, so the branch probabilities must be multiplied.
Forgetting complements such as $P(A^c)=1-P(A)$ is wrong because the tree diagram must include all possible branches from each stage.
Rounding too early is wrong because small rounding errors can noticeably change the final Bayes' theorem result.

Practice Questions

1 A disease affects $2\%$ of a population. A test is positive for $95\%$ of people with the disease and positive for $4\%$ of people without it. Find $P(\text{disease}\mid \text{positive})$ .
2 A box has two machines producing parts. Machine $A$ makes $60\%$ of the parts and has a defect rate of $3\%$ . Machine $B$ makes $40\%$ of the parts and has a defect rate of $7\%$ . Find $P(A\mid \text{defective})$ .
3 A student either studies with probability $0.70$ or does not study with probability $0.30$ . If the student studies, the probability of passing is $0.90$ ; if not, it is $0.50$ . Find $P(\text{studied}\mid \text{passed})$ .
4 Explain why a tree diagram helps identify the correct denominator when using Bayes' theorem.

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