High School: Statistics and Probability
Using Probability to Make Decisions HSS-MD.A.2
2. Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
You know how there are about a zillion names for the same drug? Or all the different names, nicknames, and scientific names for the same animal? Or that there are over twenty different synonyms for "buttocks" in the English language?
Well, students should know that mathematicians are just as good at making up words (though none as funny as "derriere" or "hiney.") So when they say expected value, they mean weighted average. But the expected value is the true mean, whereas the actual mean may vary. You know what we mean?
No? Well let's get to the "bottom" of this.
Students can think about it this way. If we conduct a short experiment with ten random coin tosses, we expect to get 5 tails. That's our expected value. However, our sample mean may actually be 3 because of the random nature of the flips combined with the small sample size. That's the real difference. Over a much longer time frame with a much large sample size, our sample mean should approach the expected value: 5 tails for every 10 tosses.
Students should know that in order to calculate our expected value, we use the formula E(x) = x1p1 + x2p2 + … + xipi, where E(x) is our expected value and p is the probability of event x occurring. Remember that x is a number associated with the event. (If you were picking coins out of a hat, the event of picking up a penny would be x = 1, while the event of picking up a dime would be x = 10.)
Bring in a deck of cards to class. Assign values for randomly drawing a card. (For example, each numbered card can equal that payoff.) Calculate the expected value of the payoff and then conduct an actual experiment to see if the actual value matches the expected value. Repeat this with increasing sample amounts to help your students understand how a sample mean may differ from the actual expected value depending on sample size.