Question

One bus arrives each day. The probability of rain in the village is 0.3. If it rains, the probability of the bus being late is 0.4. If it does not rain, the probability of the bus being late is 0.15. Create a tree diagram to model this chance process.

Answer 1

The probability of the bus being late is 0.225 or 22.5%.

Certainly! A tree diagram can be a useful visual representation for modeling probability scenarios. In this case, we can create a tree diagram to represent the arrival of the bus based on the weather conditions. Here it is:

/ \

/ \

Rain No Rain

/ \ / \

/ \ / \

0.3 / \ 0.7/ \

/ \ / \

Late On Time Late On Time

0.4 0.6 0.15 0.85

In the diagram:

- The first level represents the weather conditions: Rain or No Rain.

- The second level represents the probabilities of each weather condition occurring: 0.3 for Rain and 0.7 for No Rain.

- The third level represents the possible outcomes for the bus arrival: Late or On Time.

- The fourth level represents the probabilities of the bus being Late or On Time given the weather conditions.

You can follow the branches to calculate the overall probability of each outcome. For example, the probability of the bus being late is the sum of the probabilities of it being late in the Rain scenario and the No Rain scenario:

`\[ P(\text{Late}) = P(\text{Rain}) * P(\text{Late}|\text{Rain}) + P(\text{No Rain}) * P(\text{Late}|\text{No Rain}) \]`

`\[ P(\text{Late}) = (0.3 * 0.4) + (0.7 * 0.15) \]`

`\[ P(\text{Late}) = 0.12 + 0.105 \]`

`\[ P(\text{Late}) = 0.225 \]`

So, the probability of the bus being late is 0.225 or 22.5%.