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