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A bag contains seven red balls, six yellow balls, and four black balls.
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If a ball is randomly selected, find the probability of it being neither red nor yellow.
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Now this question is all about complementary events, and that is events whose probabilities add up to one.
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You can either have one thing happening or the other thing happening.
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Now given that we’ve got red and yellow and black balls in the bag, if we pick a ball out at random, it’s going to either be black, or red or yellow.
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We’re absolutely certain that it’s gonna be one of those two things.
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If we add up how many balls we’ve got in the bag, that’s seven plus six plus four, that’s 17 in total.
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Four of them are black, and the other 13 are either red or yellow.
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So the probability of picking a black ball is four out of 17, and the probability of picking a red or yellow ball is 13 out of 17.
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And indeed four seventeenths and thirteen seventeenths add together to make one seventeen seventeenths.
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And the question said: Find the probability of it being neither red nor yellow.
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And if we- neither red nor yellow, we’re not in this situation; we must be in this situation.
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So that’s equal to the probability of it being black which is four seventeenths.
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So our answer is four seventeenths.
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Now just before we go, let’s look at another way of doing this question.
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Now another way of looking at this is that we could either pick a red or yellow ball, or we could not pick a red or yellow ball.
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In other words, neither red nor yellow ball.
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That’s an either-or situation.
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We know one of those two things must happen, so their probabilities must add up to one.
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So we know that if the probability of picking red or yellow is thirteen seventeenths, I can see what do I have to add to that number to make one to find out the probability that it’s neither red nor yellow.
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And in fact, that’s four seventeenths, the same answer.
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And essentially that second approach is saying that something either will or won’t happen.
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We’re absolutely certain of that.
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So the probability that it’ll happen plus the probability that it doesn’t happen is equal to one.
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By rearranging that, if we know the probability that it will happen, we can work out the probability that it won’t happen.
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And if we know the probability that it won’t happen, then we can work out the probability that it will happen.