### Video Transcript

The tree diagram below shows the likelihood that it rains or does not rain and whether students walk or do not walk to school. Find the probability that a student walks to school. Find the probability that it rains given that a student walks to school.

We will begin by using the tree diagram to answer the first part of this question. We can see from the tree diagram that the weather and whether the student walks or does not walk to school are dependent. Whether it rains or does not rain does impact whether the student walks or does not walk to school.

In this part of the question, we are interested in the two paths: when it rains and the student walks and when it doesn’t rain and the student walks. We know that when dealing with a sequence of events, we multiply along the branches. This means that the probability of it raining and the student walking to school is equal to two-fifths multiplied by one-sixth. This is equal to two-thirtieths, which simplifies to one-fifteenth.

Likewise, we can calculate the probability it does not rain and the student walks to school by multiplying three-fifths by two-thirds. This is equal to six-fifteenths, which once again can be simplified this time to two-fifths.

As the student can walk to school either when it rains or does not rain, we need to add up these probabilities. The probability that the student walks to school is equal to one-fifteenth plus six-fifteenths. This is equal to seven-fifteenths.

We will now clear some space and consider the second part of this question. We recall that the second part of our question asked us to find the probability that it rains given that a student walks to school. The words “given that” mean that we are dealing with conditional probability. This can be written as the probability of 𝐴 given that 𝐵 occurs.

In this question, we are interested in the probability that it rains given that the student walks to school. As the two events are dependent, we recall the formula that the probability of 𝐴 given 𝐵 is equal to the probability of 𝐴 intersection 𝐵 divided by the probability of 𝐵. We need to divide the probability that it rains and the student walks to school by the probability that they walk. From the tree diagram, we see that the probability that it rains and the student walks is one-fifteenth. And in the first part of our question, we found that the probability that a student walks to school is seven-fifteenths. One-fifteenth divided by seven-fifteenths is equal to one-seventh.

The probability that it rains given that a student walks to school is one-seventh.