### 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 weβre calculating a
probability, we can recall that to find the probability of an event π΄, we calculate
π brackets π΄ divided by π brackets π, where π brackets π΄ is the number of
elements in event π΄ and π brackets π is the number of elements in sample space
π. Letβs have a closer look at this
question then. Weβre 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. Weβre not told the number of red
balls, but we can use a variable such as π₯ to represent this. The total number of balls in the
bag can be written as the number of white balls, thatβs 24, plus the number of red
balls, which weβve defined as π₯. The total number of balls in the
bag will be the number of elements in the sample space. So we can say that π brackets π
is equal to 24 plus π₯.

Next, we can have a look at the
probabilities. And weβre 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 weβll see if we have enough
information to solve to find the value of π₯. To calculate the probability of
getting a red ball, well, the number of elements in any event π΄ 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 π 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 π₯ over 24 plus π₯.

Now, we can take the cross product
and solve this for π₯. We then distribute seven across the
parentheses, which gives us 168 plus seven π₯ is equal to 31π₯. Then we can subtract seven π₯ from
both sides. Finally, dividing both sides by 24,
we find that seven is equal to π₯, and so π₯ is equal to seven. Remember that we defined the number
of red balls to be π₯, and so weβve worked out that the number of red balls must be
equal to seven. It can be very tempting to stop
here. But remember, weβre asked how many
balls there are in total in the bag. Weβve 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 π₯, 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.