# Video: Determining the Theoretical Probability of an Event in Marble Problems

Tim Burnham

A bag contains 7 red balls, 6 yellow balls, and 4 black balls. If a ball is randomly selected, find the probability of it being NEITHER red NOR yellow.

02:41

### Video Transcript

A bag contains seven red balls, six yellow balls, and four black balls. If a ball is randomly selected, find the probability of it being neither red nor yellow.

Now this question is all about complementary events, and that is events whose probabilities add up to one. You can either have one thing happening or the other thing happening. 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. We’re absolutely certain that it’s gonna be one of those two things.

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. Four of them are black, and the other 13 are either red or yellow. 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. And indeed four seventeenths and thirteen seventeenths add together to make one seventeen seventeenths.

And the question said: Find the probability of it being neither red nor yellow. And if we- neither red nor yellow, we’re not in this situation; we must be in this situation. So that’s equal to the probability of it being black which is four seventeenths. So our answer is four seventeenths.

Now just before we go, let’s look at another way of doing this question. 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. In other words, neither red nor yellow ball. That’s an either-or situation. We know one of those two things must happen, so their probabilities must add up to one. 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. And in fact, that’s four seventeenths, the same answer.

And essentially that second approach is saying that something either will or won’t happen. We’re absolutely certain of that. So the probability that it’ll happen plus the probability that it doesn’t happen is equal to one. By rearranging that, if we know the probability that it will happen, we can work out the probability that it won’t happen. And if we know the probability that it won’t happen, then we can work out the probability that it will happen.

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