8.4 Tree Diagrams for Conditional Probability

A conditional probability tree diagram is a visual tool used to represent and calculate the probabilities of sequential events where the outcome of one event influences the probability of the next. These are known as dependent events. Tree diagrams are particularly useful for breaking down complex scenarios into simpler, manageable steps, making it easier to understand the flow of probabilities.

Key Formula for Sequential Events

The fundamental formula used in conjunction with tree diagrams for calculating the probability of two or more dependent events occurring in sequence is the multiplication rule for conditional probability:

$$ P(A \text{ and } B) = P(A) \times P(B | A) $$

Where:

• $P(A)$: The probability of the first event ($A$) occurring.
• $P(B | A)$: The conditional probability of the second event ($B$) occurring, given that the first event ($A$) has already occurred. This is read as "the probability of B given A".
• $P(A \text{ and } B)$: The joint probability of both event $A$ and event $B$ occurring in that specific sequence.

This formula can be extended to more than two events, such as $P(A \text{ and } B \text{ and } C) = P(A) \times P(B | A) \times P(C | A \text{ and } B)$.

Example: Weather Forecast and Umbrella Choice

Let's illustrate with a practical example: determining the probability of carrying an umbrella based on the weather.

Scenario

Assume the following probabilities:

• Probability of Rain: $P(\text{Rain}) = 0.7$
• Probability of No Rain: $P(\text{No Rain}) = 0.3$

Now, consider the probability of carrying an umbrella depending on the weather:

• Given it rains, the probability of carrying an umbrella: $P(\text{Umbrella | Rain}) = 0.9$

• Given it rains, the probability of not carrying an umbrella: $P(\text{No Umbrella | Rain}) = 1 - 0.9 = 0.1$

• Given it does not rain, the probability of carrying an umbrella: $P(\text{Umbrella | No Rain}) = 0.2$

• Given it does not rain, the probability of not carrying an umbrella: $P(\text{No Umbrella | No Rain}) = 1 - 0.2 = 0.8$

Constructing the Tree Diagram

A tree diagram visually represents these probabilities as branches originating from a starting point.

1. First Level Branches: Represent the initial event (whether it rains or not).

   • Branch 1: Rain (with probability 0.7)
   • Branch 2: No Rain (with probability 0.3)

2. Second Level Branches: For each branch from the first level, represent the subsequent event (carrying an umbrella or not).

   • From "Rain":
     • Carry Umbrella (with probability $P(\text{Umbrella | Rain}) = 0.9$)
     • No Umbrella (with probability $P(\text{No Umbrella | Rain}) = 0.1$)
   • From "No Rain":
     • Carry Umbrella (with probability $P(\text{Umbrella | No Rain}) = 0.2$)
     • No Umbrella (with probability $P(\text{No Umbrella | No Rain}) = 0.8$)

Step-by-Step Calculations

To find the probability of each possible outcome (e.g., it rains and you carry an umbrella), we multiply the probabilities along each path from the root to a terminal node:

• Scenario 1: Rain and Umbrella $P(\text{Rain and Umbrella}) = P(\text{Rain}) \times P(\text{Umbrella | Rain})$ $P(\text{Rain and Umbrella}) = 0.7 \times 0.9 = 0.63$

• Scenario 2: Rain and No Umbrella $P(\text{Rain and No Umbrella}) = P(\text{Rain}) \times P(\text{No Umbrella | Rain})$ $P(\text{Rain and No Umbrella}) = 0.7 \times 0.1 = 0.07$

• Scenario 3: No Rain and Umbrella $P(\text{No Rain and Umbrella}) = P(\text{No Rain}) \times P(\text{Umbrella | No Rain})$ $P(\text{No Rain and Umbrella}) = 0.3 \times 0.2 = 0.06$

• Scenario 4: No Rain and No Umbrella $P(\text{No Rain and No Umbrella}) = P(\text{No Rain}) \times P(\text{No Umbrella | No Rain})$ $P(\text{No Rain and No Umbrella}) = 0.3 \times 0.8 = 0.24$

Final Result Verification

The sum of all probabilities calculated for the terminal nodes should equal 1, confirming that all possible outcomes have been accounted for: $0.63 + 0.07 + 0.06 + 0.24 = 1.00$

For instance, the probability of the specific outcome "it does not rain, and you do not carry an umbrella" is $0.24$, meaning there's a 24% chance of this occurring.

When to Use Tree Diagrams

Conditional probability tree diagrams are most effective when:

• Events occur in a sequence: There is a clear chronological order to the events.
• Outcomes are dependent: The probability of a later event changes based on the outcome of an earlier event.
• A clear breakdown is needed: You require a visual representation and calculation of all possible joint probabilities and conditional probabilities in a sequential process.
• Multiple stages of decisions or events: The diagram helps manage complexity when there are several steps or choices.