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.
- 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?
- 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.
- 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.
- (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.