Identify the Stages: Determine how many sequential events are occurring and the possible outcomes for each.
Assign Probabilities: Place the probability of each outcome on its corresponding branch. Ensure that for each node, the probabilities sum to .
Calculate Path Probabilities: Multiply the values along each complete path from left to right to find the probability of that specific combination of outcomes.
Sum Relevant Outcomes: If the question asks for a condition that can be met in multiple ways (e.g., 'at least one success'), add the probabilities of all paths that meet that criteria.
| Feature | Independent Events | Dependent Events |
|---|---|---|
| Probability Change | Probabilities remain constant across stages. | Probabilities in later stages change based on previous results. |
| Example Scenario | Flipping a coin multiple times. | Drawing cards from a deck without replacement. |
| Branch Values | $P(B | A) = P(B |
The Complement Shortcut: For 'at least one' questions, it is often faster to calculate rather than summing multiple paths.
Fractional Precision: Keep probabilities as fractions until the final step to avoid rounding errors that can compound during multiplication.
Verification: Always check that the sum of all final path probabilities equals . If it does not, a branch was likely missed or a calculation is incorrect.
Labeling: Clearly label the end of each path with the combined outcome (e.g., , ) to ensure you don't miss any relevant scenarios when adding.
Forgetting to Update Probabilities: In 'without replacement' scenarios, students often forget to reduce both the numerator and the denominator for the second stage.
Adding Along Branches: A common error is adding probabilities along a path instead of multiplying them. Remember: 'AND' means multiply; 'OR' means add.
Incomplete Trees: Failing to draw all branches can lead to missing outcomes. Even if an outcome isn't 'successful' for the problem, it must be included to ensure the node sums to .
