### Video Transcript

A number 𝑥 is selected at random
from the numbers one, two, three, and four. Another number 𝑦 is selected at
random from the numbers one, four, nine, and 16. Find the probability that the
product of 𝑥 and 𝑦 is less than 16.

Okay, so there are four possible
outcomes for the value of 𝑥 picked. One, two, three, and four are all
equally likely to be selected. But independently of randomly
selecting a value of 𝑥, a random value of 𝑦 is selected from the numbers one,
four, nine, and 16. Each of these values of 𝑦 is
equally likely to occur..

There are four possible values of
𝑥 and four possible values of 𝑦. And therefore, there are four times
four, which is 16 possible pairs of values of 𝑥 and 𝑦. Our task is to find the probability
that the product of 𝑥 and 𝑦 is less than 16.

When dealing with equally likely
outcomes, the probability of an event is a fraction, where the denominator is the
total number of equally likely outcomes and the numerator is the number of those
equally likely outcomes which are favorable; that is, which are part of the
event.

In our case, the event that we’re
interested in is the event that the product of 𝑥 and 𝑦 is less than 16. What is the total number of equally
likely outcomes? Well, we’ve seen it’s 16. There are 16 equally likely pairs
of 𝑥 and 𝑦.

How many of those pairs are
favorable; that is, for how many of those pairs is the product of 𝑥 and 𝑦 less
than 16? Well, let’s count them. For each pair of 𝑥 and 𝑦, we find
their product. The product of one and one is one,
the product of two and one is two, the product of three and one is three, and the
product of four and one is four.

We continue with the next row. The product of one and four is
four; of two and four is eight; of three and four is 12; and of four and four is
16. One and nine is nine. Two nines are 18. Three nines are 27. And four nines are 36. And finally, one 16 is 16, two are
32, three are 48, and four are 64. So these are the products you get
from the 16 equally likely pairs of 𝑥 and 𝑦.

Now, how many of them are less than
16? Let’s count them. One is less than 16 as are two,
three, four, four, eight, and 12. However, 16 itself is not less than
16. So we don’t count that. Nine, however, is less than 16. But the rest of the products are
all greater than or equal to 16 and so are not counted.

So how many favorable outcomes do
we have? Well, we have eight. There are eight equally likely
pairs of 𝑥 and 𝑦 for which their products 𝑥 times 𝑦 is less than 16. All this left to do is to simplify
this fraction. Eight over 16 is one-half.

So when a number 𝑥 is selected at
random from the numbers one, two, three, and four and another number 𝑦 is selected
at random from the numbers one, four, nine, and 16, the probability that the product
of 𝑥 and 𝑦 is less than 16 is a half.