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 with .
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 to find the probability of a cause after seeing an outcome. The denominator often comes from adding all paths that lead to event .
Tree diagrams make each path visible, which helps students choose the correct numerator and denominator.
Key Facts
- Conditional probability is written as , where .
- Bayes' theorem is .
- The total probability formula is for two complementary cases.
- In a tree diagram, multiply probabilities along one complete path to find a joint probability such as .
- 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 .
- The numerator in Bayes' theorem is the joint probability of the condition and the event you want, such as .
- The denominator in Bayes' theorem is the total probability of the given condition, such as .
Vocabulary
- Bayes' theorem
- A formula for finding a reverse conditional probability using .
- Conditional probability
- The probability that event happens given that event has already happened, written .
- 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 .
- Complement
- The event that does not happen, written , with probability .
- 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 with 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 must include every path that leads to event .
- 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 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 of a population. A test is positive for of people with the disease and positive for of people without it. Find .
- 2 A box has two machines producing parts. Machine makes of the parts and has a defect rate of . Machine makes of the parts and has a defect rate of . Find .
- 3 A student either studies with probability or does not study with probability . If the student studies, the probability of passing is ; if not, it is . Find .
- 4 Explain why a tree diagram helps identify the correct denominator when using Bayes' theorem.
Understanding Bayes' Theorem with Tree Diagrams
A reverse conditional probability is difficult because the evidence may have several possible causes. A positive medical test can come from a person with a disease, yet it can also come from a healthy person when the test gives a false positive. The chance that a person has the disease after a positive result depends on how common the disease was before testing.
This starting chance is called the base rate. A very accurate test can still produce many false positives if it is used on a huge group where the condition is rare. Bayes' theorem keeps the base rate in the calculation instead of focusing only on the test accuracy.
A useful way to understand this is to imagine real numbers of people rather than only decimals. Suppose one thousand people are screened. First split them into those with the condition and those without it.
Then split each group by test result. The final boxes show actual expected counts, such as people who have the condition and test positive. To find the chance that a positive person has the condition, compare the number in that one box with the total number of positive results.
This frequency method often makes the denominator feel more natural. It represents the whole group that matches the evidence being given.
Tree diagrams require careful reading because branch labels change meaning at each stage. The first branches usually show the original groups, such as trained and untrained workers. The next branches show outcomes within each original group, such as passing or failing an inspection.
A probability on a second branch is not usually a probability for everyone. It is a probability only within the group at the start of that branch. This is why branch probabilities coming from different starting points should not be compared carelessly.
Check that the branches leaving any one point add to one. Check that the probabilities at the ends of all complete paths add to one too.
Students often make an error by using the probability of a positive result among people with a condition as though it were the probability of the condition among people with a positive result. These statements can have very different values. Write the target group in words before calculating.
For example, the target group might be all people who tested positive. Then identify every path that belongs in that group. Another common mistake is forgetting a possible cause of the evidence.
If a result can occur for three different groups, all three paths affect the total. In exam questions, underline words such as given, among, after, and knowing that.
They signal which event is already known and which event must be found. This habit helps prevent reversing the condition without noticing.