Does It Matter Which Winner You Saw?   

Scenario: Students at your school have just finished competing in the qualifying round of a nationally sponsored contest on mathematical reasoning and sense making. When the work was scored, it turned out that four students at your school all had perfect preliminary papers—two girls and two boys. The school decided to hold a random drawing among these four students to select two of them to send to the national finals. The drawing takes place in the school auditorium. You show up late to the drawing, just as one of the winners—a girl—is leaving the stage amid cheers.

1. Suppose that the girl that you saw leaving the stage is the first winner. What is the probability that the second winner will also be a girl?
2. Suppose that the girl that you saw leaving the stage was the second winner. What is the probability that the first winner was also a girl?

Remember: it is important for you and your students to share your reasoning!

Solution: 

Two schools of thought frequently emerge in reasoning about the “Which Winner” problem.

1. If the girl that you saw on the stage was the first winner, then out of the three students left for the random drawing for a second winner, only one is a girl, and two are boys. The chances that the second one is also a girl will be 1 in 3, or 1/3.

However, if the girl that you saw on stage was actually the second winner, then the chance that the first winner was also a girl is 1/2, because before the first girl was picked, there were two girls and two boys, so the chance of a choosing a girl was then 2 in 4, or 1/2. 

2. (Same as above) If the girl that you saw on the stage was the first winner, then out of the three remaining students, one is a girl and two are boys, so the chances that the second winner will also be a girl is 1/3.

(Different than above) It doesn’t make any difference whether you saw the first girl or the second girl; the fact that you saw a girl winning at all means that there is only a 1 in 3 chance that the other winner was a girl, so, the probability that the first winner was also a girl if the girl that you see on the stage is the second winner is also 1/3.

You may have encountered one or both of these lines of reasoning in discussions with your students—or among your colleagues. One way to try to resolve this debate—is the probability 1/2 or is it 1/3 that the first winner was also a girl—is to simulate a drawing for the two winners. Using four objects–for example, two green and two yellow chips—and a bag, simulate, say, 50 drawings of two chips. Let green (G) be boys and yellow (Y) be girls. (You might also create this simulation by using a number of probability software applications). Draw a chip and record the color, then draw a second chip (from the remaining three—no one can win twice!—and record its color. You will have 50 pairs of results—for example: YY, YG, and so on. Circle only those pairs where the second winner is a girl. Looking at the circled pairs, determine what percentage of those pairs also have a girl as a first winner. This simulation will provide an estimate for the probability that the first winner was also a girl, given that you saw that the second winner was a girl.

The “Which Winner” problem is a nice example of conditional probability. You actually have some information when you know that the second winner is a girl, and such conditional information can influence probabilities.

Finally, here is an extension of the “Which Winner” problem, as posed by one of our e-Summing Up readers:

You walk in after the ceremony is over, and you say to your colleague, "Just tell me yes or no—was at least one of the winners a girl?" You are told yes. So, what is the probability that both winners were girls?

An Additional Note:

The solution shown below for the June problem was submitted by a student in grade 8.

If first winner was a girl,                                If second winner was a girl: 

BB                                                                    BB 

GG                                                                   GG 

chances are 1/3 the second one                     chances are 1/3 the first one  

will also be a girl                                                        will also be a girl 

According to this student, it doesn’t make any difference whether the girl that you saw on the stage was the first winner or the second winner. The probability of the other winner also being a girl is 1:3 in either case. It’s hard to argue with this reasoning.