Probability Tree Diagrams for Dependent Events

Understanding Probability Tree Diagrams for Dependent Events

• Dependent events are events where the outcome of the first event influences the outcome of the second event.
• A Probability Tree Diagram for dependent events visually represents the different possibilities and their respective probabilities.
• The branching structure of the diagram allows for clear understanding of the sequential progression of events, especially when the events are dependent.

Constructing Probability Tree Diagrams for Dependent Events

• Identify the first event and its possible outcomes. Draw a dot or a node to represent the start of the event.
• Create branches for each outcome of the first event. Write probabilities on these branches. Remember that for a single event, the total of the probabilities should total to 1.
• Determine the possible outcomes of the second event after each outcome of the first event. These are the dependent outcomes.
• From the end of each first event branch, draw new branches for each dependent outcome of the second event. The probabilities must reflect the dependency as a consequence of the first event.
• Continue the process for subsequent events, each time branching out from all endpoints from the preceding event.

Interpreting Probability Tree Diagrams for Dependent Events

• Each pathway from the start of the diagram to an endpoint represents a unique sequence of outcomes.
• The probability of each sequence of outcomes is calculated by multiplying the probabilities along the path.
• Note, the probabilities on the branches of a dependent event may vary, unlike those of independent events, because they are influenced by the outcomes of the previous events.
• Observe the changes in branch probabilities as this shows the influence one event has on the next, an essential characteristic of dependent events.

Problem Solving with Probability Tree Diagrams for Dependent Events

• To find the probability of a specific sequence of outcomes, locate the pathway on the diagram that represents these outcomes and then multiply the probabilities along this path.
• To find the aggregate probability of multiple sequences, sum up the obtained probabilities of those sequences.
• Always confirm that the total probability at the end, when all outcomes are considered, sums to 1. If it doesn’t, it’s likely an outcome has been overlooked.
• Application of conditional probability can also be understood using these diagrams. This is when the probability of an event, given that another event has occurred, is being calculated. It is especially evident in tree diagrams representing dependent events.