Probability Trees

With more complicated probability questions, drawing a probability tree can be a really useful way to visualise the potential outcomes.

Work through this example to see how to use probability trees in your work.

Example

The probability that it will rain on any particular day in September is 0.3. If it is raining, the probability that the bus will be late is 0.6. If it is dry, the probability that the bus will be late is 0.2.

Display this information on a probability tree.

Step 1: Sketch the structure of the probability tree.

Step 2: Add in the probabilities.

Questions: Using the probability tree

Question 1. What is the probability that it will be raining, and the bus will be on time?

Think about the branches like paths. Start on the left, and choose the correct paths as required in the question. Make a note of the probabilities on the branches that you select.

As this question is asking for the probability that it’s raining AND the bus is on time, we are going to use the ‘AND’ rule, and multiply the probabilities to find the answer:

  • 0.3 \times 0.4 = 0.12

The probability that it will be raining, and the bus will be on time is 0.12.

Question 2: What is the probability that the bus will be on time, regardless of the weather?

This time, there are two possible paths that we can take through the probability tree. Either it could be raining and the bus is on time, or it could be dry and the bus is on time.

Work out each path separately:

  • Yellow path (raining, on time) = 0.3 \times 0.4 = 0.12
  • Green path (dry, on time) = 0.7 \times 0.8 = 0.56

Now, we must add these two probabilities together. This is because either it could be raining and on time, OR dry and on time. We must use the ‘OR’ rule, and add them together:

0.12 + 0.56 = 0.68

The probability that the bus will be on time, regardless of the weather is 0.68.

Question 3: What is the probability that the bus will be late, regardless of the weather?

We can use our previous answer to help answer this one easily. This question is the opposite of the previous one. As there are only two possibilities (bus is late, or on time) we can take the probability away from 1 to find the probability that the bus is late:

1 - 0.68 (probability that the bus is on time) = 0.32 (probability that the bus is late)

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