### 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.