IPS - Day 14

Today we continued exploring calculating probabilities and running simulations. We looked at the birthday problem with the intent of determining the probability.

The first task was to simulate the situation. I had students consider the scenario and then we discussed their ideas. The class was in general agreement that they could generate values from 0-364 or 1-365, do this 40 times and look for duplicates in their lists. The point of confusion came about in what data was to be collected from this simulation.

The question of interest was "What is the probability that at least two people have the same birthday in a room of 40 people?" In performing a simulation, the focus should then be on either the room does or doesn't contain at least one match. In fact, we don't care if there are multiple matches, just as long as there is one.

Students, on the other hand, wanted to focus on the number of matches that they had. Despite discussing the issue and explicitly directing them to record the number of trials that had matches, many students continued to count total matches in their 40 trials. It took some individual discussions to get everyone reporting the same data.

We generated 104 simulations and had 90 occurrences of two or more matches. This translated to a probability of 86.5%. I did wonder about one group's data as it showed a much smaller match rate than everyone else (9 out of 15). This one group accounted for almost half of the misses and although I did not analyze the results statistically to affirm consistency, the data appears suspect on the surface.

The 86.5% result is on par with the theoretical result of approximately 90%. In order for students to see this they were asked to calculate the theoretical probability. This calculation takes a lot of thought and can be quite involved. I like to present the challenge, not with the expectation that students will be able to determine how to calculate the probability but to help them make connections as to how basic probability rules can be applied in complex situations.

Students were stuck as to how to proceed. I asked two students to the front of the class and asked what is the probability that their two birthday's match? Students were quick to respond 1/365. I then asked another student to join the first two. I asked what is the probability that at least two of these students have the same birthday? This caused students to pause.

Say we have students A, B, and C. We then must consider that AB match, AC match, BC match, and ABC match. This starts to become a more complicated endeavor. If we add a fourth person, D, then there are even more combinations of birthday matches to consider. Continuing to 40 people seems beyond possible to calculate.

I have told the class before, and this problem is a great example, that when calculating probabilities and the calculations start to get messy and unmanageable, they should look at calculating the complement of probability.

Students are still not comfortable with the idea that P(A) + P(A^C) = 1. In this case P(A^C) is the probability that no one has the same birthday. This probability is a much more straightforward calculation than P(A), the probability that at least 2 people have the same birthday.

We proceeded sequential, the probability that the first two people do not have the same birthday is 364/365. Another person is added to the room. The probability this person's birthday does not match either birthday is 363/365. We continue, each time adding a person and subtracting an available day from the numerator. The result is
               P(A^C) = 364/365 x 363/365 x 362/365 x 361/365 x ... x 325/365

And the P(A) is simply 1 - P(A^C).

This calculation still takes some work. I showed students how to perform this calculation on their calculator using list operations and list math functions. Typically students do not know more than simple basics on how to use their graphing calculators. I want to show them that they can perform complex operations with just a little effort. At the same time, I probably need to rethink how to best go through this. I am thinking a step-by-step investigation worksheet may work better. Students would probably be more willing to follow the steps and answer checking questions along the way. There's always something new to add for the next semester.

I concluded with having students summarize their work and writing down two questions they still have about probability and simulations.

Visit the class summary for a student's perspective and to view the lesson slides.