IPS - Day 23

Today we continued looking at probability models and expected values. Many students did not attempt to complete the Shooting Free Throw worksheet. I let it be known, in no uncertain terms that when I give an assignment that I expect it to be completed.

Because of the issues with this I decided to skip over the simulation portion of shooting two free-throws and asked students to focus on looking at the problem from a theoretical perspective. In this case the struggles were with recognizing that scoring one point could occur in two different ways: a miss and a make or a make and a miss.

Students would ask if their result was correct and I would ask if their probabilities summed to one. When they realized they didn't it became an issue of why not? They then discovered that they hadn't accounted for all of the outcomes.

With the probability model constructed I asked students to calculate the expected value. I had to refer back to the One Girl Family Planning worksheet and its definition of expected value. Students calculated the expected value. We briefly discussed the meaning of expected value.

I then had students look at the simple dice game that I had used a couple of days ago. In this game, you roll the die and break even with a roll of 1, win $5 with a roll of 2 or 3, and lose $5 with a roll of 4, 5, or 6. I asked students to calculate the expected value for this game. I wanted to use this as a check to see if students knew the basic mechanics for calculating an expected value. I was pleased to see that a large percentage of the class was able to calculate the expected value of -$0.83 correctly.

I asked what the expected value meant. Students said it meant that you would lose money when playing the game. I tried to push their thinking further. I explained that the expected value was the amount I could expect to lose on average. For example, if I played one million times, I told students that on average I would have lost $0.83 cents. I then asked them how much total money would I have lost? This puzzled them for a few minutes. I said that after playing the game one million times my average loss was $0.83 cents. Students started working through and figured that I would lose $830,000 in total. I thought this was a good way to illustrate how the expected value represents an average of the results that will occur but the cumulative impact can be much different.

We then started to look at the 39-Game Hitting Streak investigation. Students struggled with how to verify a given probability. I told them to consider from a theoretical perspective the events that take place to reach the stated outcome.

The investigation asks students to consider the probability of hitting streaks of 1, 2, and 3 games. The tendency is to focus on the probability of getting at least one hit in a game and stopping. The issue is that a hitting streak of a given length means the streak ends. For a one-game hitting streak, a batter must get at least one hit in the first game and then get no hits in the second. For a two-game hitting streak, a batter must get a least one hit in games one and two and then not get a hit in game three. This pattern continues for whatever length hitting streak you may want to consider.

Students were just starting to understand the issue that the hitting streak must end when we got to the end of the class. We'll finish up this investigation on Monday and then take a look at some other situations involving expected values.

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