Probability Tree Diagrams

• The Sum Rule for Branches: At any given node, the sum of the probabilities of all branches extending from that node must equal exactly $1$. This is because the branches represent a set of mutually exclusive and collectively exhaustive outcomes for that specific stage of the experiment.

• The Multiplication Rule (AND): To find the probability of a specific sequence of events (a path), you must multiply the probabilities along the branches of that path. This follows the formula $P(A \cap B) = P(A) \times P(B|A)$, where $P(B|A)$ is the probability of $B$ occurring given that $A$ has already happened.

• The Addition Rule (OR): If an event can occur through multiple different pathways, the total probability of that event is the sum of the probabilities of each individual pathway. This applies when outcomes are mutually exclusive, meaning only one path can be realized in a single trial.

Independent vs. Dependent Events

• Independent Events: In these scenarios, the outcome of the first event does not affect the probability of the second. The probabilities on the second set of branches remain identical regardless of which first branch was taken.
• Dependent Events: These often occur in 'without replacement' scenarios. The probability of the second event changes based on the first outcome, requiring the tree diagram to show different values on the second-stage branches.

• The 'Sum to One' Check: Always verify that the probabilities on any group of branches originating from the same node add up to $1$. If they do not, there is either a calculation error or a missing outcome.

• Final Probability Verification: After calculating the probabilities for all possible paths, sum them all together. The total sum of all final path probabilities must equal $1$, representing the entire sample space.

• Read the Wording Carefully: Look for phrases like 'at least one' or 'exactly two'. For 'at least one', it is often faster to calculate $1 - P(\text{none})$ rather than summing multiple positive outcomes.

• Fraction vs. Decimal: Keep probabilities as fractions during the calculation process to maintain precision. Only convert to decimals at the final step if the question specifically requests it.

• Adding Instead of Multiplying: A frequent error is adding probabilities along a path instead of multiplying them. Remember that a path represents 'Event 1 AND Event 2', which requires multiplication.

• Static Probabilities in Dependent Scenarios: Students often forget to reduce the denominator (and sometimes the numerator) when dealing with 'without replacement' problems. This leads to treating dependent events as independent.

• Incomplete Trees: Failing to draw all possible branches can lead to an incomplete sample space. Even if an outcome seems irrelevant to the specific question, drawing it helps ensure the total probability sums to $1$.