Step 1: Identify Stages: Determine how many events are happening in sequence and how many outcomes each event has.
Step 2: Label Branches: Assign probabilities to each branch. If the events are dependent (like sampling without replacement), ensure the probabilities in the second stage reflect the outcome of the first.
Step 3: Calculate Path Probabilities: Multiply across each horizontal path to find the probability of that specific combined outcome.
Step 4: Sum for Results: Identify which paths satisfy the condition of the question and add their calculated probabilities together.
| Feature | Independent Events | Dependent Events |
|---|---|---|
| Definition | Outcome of first does not affect the second | Outcome of first changes the probability of the second |
| Branch Values | Probabilities remain identical across all sets of branches | Probabilities change based on the previous branch taken |
| Example | Flipping a coin twice | Drawing two cards without replacement |
The '1 - P(None)' Shortcut: When asked for the probability of 'at least one' event occurring, it is often faster to calculate the probability of the event never occurring and subtracting it from 1.
Verification: Always check that the sum of all final path probabilities equals 1. If it does not, there is an error in your branch calculations or you have missed a path.
Fraction Management: Keep probabilities as fractions throughout the calculation to maintain precision; only convert to decimals at the final step if requested.
Forgetting to Update Denominators: In 'without replacement' scenarios, students often forget to reduce both the numerator and the denominator for the second stage branches.
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 that are necessary for the final 'OR' calculation.
