Question Video: Finding the Probability Distribution for a Random Variable from a Context Mathematics

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 (e.g., 𝑋 = 2 if the top ranked student is a boy and the second ranked student is a girl).

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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.

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