Step 1: Identify Stages: Determine how many events are occurring in sequence. Each event corresponds to a 'column' of branches in the diagram.
Step 2: Label Probabilities: Assign the probability of each outcome to its respective branch. Ensure that for each split, the probabilities sum to 1.
Step 3: Calculate Path Probabilities: Multiply the values along each unique path from left to right to find the probability of that specific combination of events.
Step 4: Sum Desired Outcomes: Identify all paths that satisfy the problem's criteria and add their calculated probabilities together to get the final answer.
Independent Events: The outcome of the first stage does not change the probabilities in the second stage. For example, flipping a coin twice or 'sampling with replacement' results in identical branch probabilities at every stage.
Dependent Events: The outcome of the first stage directly alters the probabilities of the second stage. This typically occurs in 'sampling without replacement' scenarios, where the total number of items decreases after the first draw.
| Feature | Independent Events | Dependent Events |
|---|---|---|
| Branch Values | Remain constant across stages | Change based on previous outcome |
| Context | With replacement / Separate objects | Without replacement / Same pool |
| Calculation | $P(B | A) = P(B)$ |
Avoid Early Simplification: When working with fractions, do not simplify them until the very end. Keeping a common denominator (e.g., in all outcomes) makes adding the final path probabilities much easier and less prone to error.
The 'At Least One' Shortcut: If asked for the probability of 'at least one' outcome occurring, it is often faster to calculate . This avoids the need to sum multiple different paths.
Verification Check: Always verify that the sum of all final path probabilities equals exactly 1. If they do not, there is likely an error in your multiplication or you have missed a possible outcome.
Labeling Clarity: Clearly label the end of each path with the combined outcome (e.g., for Head then Tail) to avoid confusion when selecting which paths to add together.
