More on Probability Models and Expected Values

Today was a wrap up of probability models and expected value. We were in a computer lab today, so I spent some time on using Minitab as students begin to gather and analyze data for their projects. I was hoping to get into normal models today as well but that will have to wait until next week.

I used roulette as an example to have students create probability models and calculate expected values. I like using this since the probability models are simple and it turns out that all of the expected values are the same, regardless of whether you be on a specific number, bet on even or odd, bet on red or black, bet on the 2nd 12 numbers, bet on the center column; every expected value turns out to be the same. This does two things: 1) it provides easy practice for students to create a probability model and calculate the expected value and 2) makes students understand that a game like roulette is designed based on probabilities and expected values. The expectation from the casino's perspective is that you will lose. This is how casinos can build ostentatious hotels and provide customers free rooms and free drinks.

After, we took a look at a specific probability model, the binomial model. This model is generated from Bernoulli trials, which we defined. A Bernoulli trial has three characteristics:

1. There are fixed probabilities.
2. There are only two possible outcomes for each trial.
3. Each trial is independent.

For a binomial model, we are examining the number of successful outcomes for Bernoulli trials for any given amount of trials. For example, the probability of tossing 7 heads in 10 coin tosses.

After examining a binomial model situation we went into how to calculate the probabilities. With the other non-topic items that were being covered today that was as much as we accomplished.

Below is an outline of today's lesson with italicized comments enclosed in square brackets [like this].

·         Roulette expected values [had a student pick a number and then used this to calculate different expected values based on the color and number picked, i.e. red, even, 1-18, etc.]

·         Coin flipping questions

   o   Have students calculate probabilities

   o   What are characteristics of these problems [posed this question and students came up with all three requirements.]

      §  Two possible outcomes

      §  Constant probability

      §  Independent trials

      §  These are Bernoulli trials

      §  Number of successes in fixed trials is binomial

   o   How did you go about calculating these probabilities?

      §  Success/failures and number of ways

      §  Give binomial formula and explain

      §  Discuss how to calculate with a calculator [Used a baseball player getting 4 at bats and possessing a .300 batting average. What is the probability of getting exactly 2 hits and what is the probability of getting at least 3 hits.]

   o   What is the mean and standard deviation for a binomial model [Asked class how many heads they would expect to toss in 10 coin tosses. Discussed how the obtained 5 and then went into formulas. Showed the standard deviation only to reinforce that a probability model possesses both a mean and a standard deviation.]

      §  np and sqrt(npq)