A group of 68 school children completed a survey asking about their television preferences. The results show that 43 of the children watch channel A, 26 watch channel B, and 12 watch both channels. If a child is selected at random from the group, what is the probability that they will watch at least one of the two channels?
Let’s visualize what’s happening here. We have the students that watch channel A. And we also have students that watch channel B. In addition to that, we have students that watch both channels A and B, which will go in this overlap. And then, we should know, in our sample space, there may be children that do not watch channel A or channel B. We know that 12 students watch both channels. And then we see that 43 children watch channel A. But we need to be careful here because the children who watch channel A will be equal to the children who watch only channel A plus the children that watch both A and B.
We know that 43 watch channel A, and we also know that 12 of those watch both channels. In order to find the value that goes in this space and number of children that only watch channel A, we’ll need to solve. We’ll subtract the number of students that watch both A and B from the students that watch channel A. 43 minus 12 is 31. So we can say 31 of the students watch only channel A plus the 12 students that watch channel A and B. And that equals the 43 children that watch channel A.
Since 26 students watch channel B, we need to follow the same procedure. The students that watch channel B will be equal to the students that watch only channel B plus the students that watch both. In order to find the number of students that only watch channel B, we’ll need to subtract 12 from 26. And we get 14. At this point, it’s good to add these values up. We have only A plus A and B plus only B. When we add those together, we get 57. And that means we could say 57 students watch A or B. Since there were 68 students that participated, we can subtract 57 from 68 and we get 11. This tells us that there are 11 students that do not watch A or B.
Since we want to know the probability that a child selected at random watches at least one of the channels, that will be the probability of A or B. This probability will be equal to the number of ways you could choose a child that watches channel A or channel B over all the options. There are 57 different children that watch channel A or B. That means you could choose any one of the 57 out of the 68 possible outcomes, which makes the probability 57 over 68. This value doesn’t reduce any further, and so it’s our final answer.