We started class looking at a roulette table. Gambling games and Las Vegas catch students' interest and it is valuable for students to learn that large, ornate hotels and casinos are paid for by the public's misconception of expected value.
I use the roulette table as an opener but, for now, it is only on an informal basis that we talk about what should be expected to happen if the game were played repeatedly. In this discussion, students felt like they would come close to breaking even.
Next was a discussion of how probability models could help in looking at what is expected. I used a simple example that helps illustrate the issues. In this example you are playing a game that costs $5 to play. I you roll a one you receive $5. If you roll a 2 or 3 you receive $10, and if you roll a 4, 5, or 6 your receive nothing. The question is what do you expect to happen if you play this game repeatedly?
I let students think about the situation and then discuss it among themselves. First, I was pleased that students readily accounted for the cost and related that the actual outcomes for the situation were $0, $5, and -$5.
The discussions tended to focus on the values of the outcomes and not to account for the frequency of each outcome. About two thirds of the class felt they would come out a head or even. The others thought they would lose. Those thinking they would win focused on the winning of $10 and that this would offset the loss of $5.
Those that thought they would lose focused on the frequency of occurrence. In this case, half the time they would lose $5. For the other half, they would not win something part of the time and therefore they would not win as much as the lost. We set up a probability model to help illustrate the issue. The class agreed that this argument made sense.
Dice Game Probability Model
| Outcome | $0 | $5 | -$5 |
| Probability | 1/6 | 2/3 | 1/2 |
We didn't get into calculating the expected value at this point. Rather, I told the class that not only could we determine if we would win or lose but we could also determine how much we could expect to lose.
With that we dove into the first of a series of investigations drawn from Navigating Through Probability, Grades 9-12. The first investigation we looked at was One Girl Family Planning. This activity builds on previous investigations done in class.
Students were comfortable with the simulation piece. The issues came in with calculating average children per household. The tendency is to either calculate average households per group or average children per group. It takes some prodding to get students to understand they need to determine the pool of children available and how many households these children are being divided up by.
The next hurdle comes with calculating theoretical probabilities. Again, some questioning helped students realize how this should be calculated. They were also able to calculate their expected values by using a quantity of 500 or 1000 to get a theoretical mean.
I was able to have a couple of students explore the idea of whether they needed the value of 500, say, to calculate the mean. These students thought that it was necessary. I asked them to describe how they performed their calculations. They basically said they multiplied all the values by 500 and at the end divided by 500. I asked them if they were multiplying everything and dividing everything by 500 if the 500 was necessary. The students started to realize that maybe the 500 value wasn't necessary. I asked them to check to see what would happen if they didn't use the 500 when calculating the mean.
The rest of the class will finish off the investigation at home and we'll discuss at the start of next class.
Visit the class summary for a student's perspective of the class and to view the lesson slides.
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