# Question Video: Using Theoretical Probability to Solve Problem Mathematics

A bag contains 24 white balls and an unknown number of red balls. The probability of choosing at random a red ball is 7/31. How many balls are there in the bag?

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### Video Transcript

A bag contains 24 white balls and an unknown number of red balls. The probability of choosing at random a red ball is seven over 31. How many balls are there in the bag?

So let’s say that we’ve got this bag which has 24 white balls. It also has an unknown number of red balls. So there could be one or two or three or even more than 24. We don’t know. What we are told, however, is that the probability of choosing a red ball is seven over 31. We can answer this question in one of two different ways, either by finding the number of white balls first or by finding the number of red balls first.

In order to find the number of white balls first, we need to remember that in any situation, the probabilities will sum to one. Because we only have white balls and red balls in the bag, then we can say that the probability of getting a white plus the probability of getting a red must be equal to one. We can rearrange this to give us that the probability of getting a white is equal to one minus the probability of getting a red ball. As we’re told that the probability of getting a red is seven over 31, then we need to work out one minus seven over 31. Since one can be written as 31 over 31, then we evaluate this as 24 over 31. So now we know that the probability of picking a white is 24 over 31.

Because this is a simple event, that is, an event with a single outcome, then we can use the fact that the probability of an event is equal to the number of possible outcomes over the total number of outcomes. We can use the information that we’ve got about the white balls. We can say that the probability of picking a white is equal to the number of white balls over the total number of balls. So filling in the information, the probability of a white ball is 24 over 31. And we were told in the question that there are 24 white balls. And we need to work out the total number of balls. So now we have this equation with two fractions that are equivalent to each other. However, since the numerators are equal to each other, they’re both 24, then the denominators must also be equal to each other, which means that the total number of balls in the bag must be 31.

Before we finish with this question, let’s have a look at the alternative method of finding the number of red balls. We can keep the same probability equation, only this time we’ll fill in the information about the red balls. We were told that the probability of a red ball is seven over 31. We don’t know the number of red balls, but we can use a little bit of algebra. And let’s define the number of red balls with the variable 𝑥. The total number of balls then will be the number of red balls, that’s 𝑥, plus the number of white balls, that’s 24.

We could then solve this by starting with the cross product. So we’d have seven multiplied by 𝑥 plus 24 is equal to 31𝑥. Distributing the seven across the parentheses would give us seven times 𝑥, and seven times 24 is 168. Subtracting seven 𝑥 from both sides, we’d have 168 is equal to 24𝑥. Then dividing both sides by 24, we’d have that seven is equal to 𝑥. Since we defined 𝑥 to be the number of red balls, then we’ve worked out that the number of red balls in this bag is seven. We didn’t just want to find the number of red balls, however; we wanted to find the total. There are 24 white balls and seven red balls. So that would give us 31 in total, which confirms the original answer.