### Video Transcript

A children’s entertainer has two
boxes. Box 𝐴 contains one red balloon,
two green balloons, and seven black balloons. Box 𝐵 contains four red balloons,
four green balloons, and two black balloons. By making a two-way table,
calculate the probability that the selected balloon is not red if it is drawn from
box 𝐵.

We’ve been told to approach this
problem by making a two-way table. This is a way of sorting data based
on two characteristics. Let’s think about the
characteristics of the data that have been given here.

The children’s entertainer has two
boxes: box 𝐴 and box 𝐵. So the first characteristic is
which box the balloon is in. In each box there are red, green,
and black balloons. So the second characteristic is the
color of the balloon. We can therefore set up a two-way
table in which the columns represent the boxes and the rows represent the color of
the balloons.

We then complete the table using
the information in the question. Box 𝐴 contains one red
balloon. Box 𝐴 also contains two green
balloons, and seven black balloons. In box 𝐵, there are four red
balloons, four green balloons, and two black balloons.

It’s useful to also add the row and
column totals, as well as the overall total. Summing down the columns, we find
that there are 10 balloons in box 𝐴 and 10 balloons in box 𝐵. And then summing across the rows,
we find that there are five red balloons in total, six green balloons in total, and
nine black balloons in total. The overall total can be found by
summing either the row or column totals, and in both cases, it gives 20.

Now that we’ve completed the
two-way table, we’re ready to calculate the required probability. The probability we’re asked for is
the probability that the selected balloon is not red if it is drawn from box 𝐵. This means that we’re not actually
interested in the balloons in box 𝐴 at all. So instead of the probability being
out of the total of 20 balloons, it will actually just be out of the 10 balloons in
box 𝐵. Of the 10 balloons in box 𝐵, a
total of six of them are not red, so the probability will be six-tenths.

We can denote this probability
using the notation for a conditional probability, which is a vertical line. We read this as the probability
that the balloon is not red, given that it was taken from box 𝐵.

Finally, we need to give this
fraction in its simplest terms, so we divide both the numerator and denominator by
two. By making a two-way table, we’ve
calculated that the probability that the selected balloon is not red if it is drawn
from box 𝐵 is three-fifths.