What Is Random Behavior:

A Develop Understanding Task

Launch: A basketball player who makes 70% of her free throws has missed four in a row. Before her next shot, someone in the stands is sure to say that she is “due” to make this one. You have lost card games to your friends every night this week; you think to yourself, “Well, next week I’m due to with a few.” A couple has three children, all boys. “The next one is due to be a girl,” a friend remarks. Each of these events, the free throw, the card game, the birth of a child, involves chance outcomes that cannot be controlled. In the long run, the basketball player makes 70% of her free throws, you may win 60% of the card games, and approximately 50% of the babies born are girls. But do these long-run facts help us to predict behavior of random events over the short run?

Can knowledge of past events help us predict the next event in a random sequence?

Explore:

The large bag of beans in the classroom contains some that are white. Beans are to be selected from the bag one at a time and laid on the desk, in the order selected. Your goal is to come up with a rule for predicting the occurrence of white beans.

  1. Take a sample of five beans from the bag to serve as data for making the first prediction.
  2. Construct a rule for predicting whether or not the next bean drawn will be white. You may use the information on the first five selections, but your rule must work for any possible sample selection of beans.
  3. Record the prediction your rule generates for whether or not the next bean will be white. Now, randomly select the next bean. Record whether or not your prediction was correct.
  4. Continue making predictions for the next selection and then checking them against the actual selection until you’ve made an additional 25 selections (beyond the original five). Calculate the percentage of correct predictions your rule generated.

Discuss:

  1. Collect the class data into a dot plot where each dot is the percentage correct for one rule. What do you observe about the plot?
  2. Compare your rule with others produced in the class. Arrive at a class consensus on the “best” rule for predicting white beans and on the “worst” rule.
  3. Test the “best” rule in the class for another 25 bean selections. Test the “worst” rule in the class for another 25 bean selections. Is the “best” rule still “best”? Is the “worst” rule still “worst”?
  4. In light of your exploration with beans today, can knowledge of past events help us predict the next event in a random sequence? Explain and justify your answer.