### Video Transcript

Two boys and two girls are ranked
according to their scores on an exam. Assume that no two scores are alike
and that all possible rankings are equally likely. Find the probability distribution
of the random variable 𝑋 expressing the highest ranking achieved by a girl, for
example, 𝑋 equals two if the top-ranked student is a boy and the second-ranked
student is a girl.

So we have four students and
they’re going to be ranked according to their scores on an exam. We need to begin by listing all the
different ways we can place these two boys and two girls in the four positions. Now notice that we don’t care which
girl or which boy is in each position, only whether the person in each rank is a boy
or a girl.

Let’s begin by putting a girl in
the first position. We can then put the other girl in
position two, position three, or position four. Each time, the remaining two gaps
must both be filled by boys. If, instead, we were to have a boy
in the first position, then in the same way, the second boy could either be in the
second position, the third position, or the fourth position. And in each case, the remaining two
positions would both be filled by girls. So we see that we have six possible
ordering of two girls and two boys. Remember, each time we’re not
interested in which boy or which girl it is. The question tells us that all
possible rankings are equally likely, which means that the probability associated
with each of these orderings is one-sixth.

The discrete random variable 𝑋
we’re interested in is the highest ranking achieved by a girl. In each of the first three cases,
there’s a girl in first place, so the value of 𝑋 is one. In the fourth case, the girls are
in third and fourth positions, so the highest ranking achieved by a girl, the value
of 𝑋, is three. In the fifth case, the girls are in
second and fourth positions, so the highest ranking is two. And in the sixth case, the girls
were in second and third positions. The highest ranking is again
two. These values in the final column
give the range of our discrete random variable 𝑋. It can take the values one, two, or
three.

We then need to fill in the
associated probabilities. Remember, the value one appears
three times, so its total probability is three times one-sixth. That’s three-sixths, which can be
simplified to one-half. The value two appears twice. So its total probability is
two-sixths, which simplifies to one-third. And finally, the value three only
appears once, so its probability is one-sixth. And so we found the probability
distribution of the random variable 𝑋. The values in the range are one,
two, and three with corresponding probabilities one-half, one-third, and
one-sixth.