Question Video: Using Theoretical Probability to Solve Problem Mathematics

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?

Because we’re calculating a probability, we can recall that to find the probability of an event 𝐴, we calculate 𝑛 brackets 𝐴 divided by 𝑛 brackets 𝑆, where 𝑛 brackets 𝐴 is the number of elements in event 𝐴 and 𝑛 brackets 𝑆 is the number of elements in sample space 𝑆. Let’s have a closer look at this question then. We’re told that there are 24 white balls in the bag, but an unknown number of red balls. We can use the same style of notation then to note that the number of white balls must be 24. We’re not told the number of red balls, but we can use a variable such as 𝑥 to represent this. The total number of balls in the bag can be written as the number of white balls, that’s 24, plus the number of red balls, which we’ve defined as 𝑥. The total number of balls in the bag will be the number of elements in the sample space. So we can say that 𝑛 brackets 𝑆 is equal to 24 plus 𝑥.

Next, we can have a look at the probabilities. And we’re told that the probability of picking a red ball is seven over 31. We can now write this probability formula in terms of finding the probability of a red ball. Then we’ll see if we have enough information to solve to find the value of 𝑥. To calculate the probability of getting a red ball, well, the number of elements in any event 𝐴 for a red ball would correspond to the number of red balls in the bag. The denominator, which is the number of elements in sample space 𝑆 in this context, is simply the total number of balls in the bag. We can then plug in the values that we have for these expressions. So we have seven over 31 is equal to 𝑥 over 24 plus 𝑥.

Now, we can take the cross product and solve this for 𝑥. We then distribute seven across the parentheses, which gives us 168 plus seven 𝑥 is equal to 31𝑥. Then we can subtract seven 𝑥 from both sides. Finally, dividing both sides by 24, we find that seven is equal to 𝑥, and so 𝑥 is equal to seven. Remember that we defined the number of red balls to be 𝑥, and so we’ve worked out that the number of red balls must be equal to seven. It can be very tempting to stop here. But remember, we’re asked how many balls there are in total in the bag. We’ve already worked out an expression for the number of balls in the bag. It was that given by this number of elements in the sample space. So the total number of balls is 24 plus 𝑥, which was seven. And when we add those together, we get the value of 31. Therefore, we can give the answer that there must be 31 balls in the bag.