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Coincidence or Causation?

Module by: Interactive Mathematics Program

Intent

In this activity, students consider whether previous experiences should be taken into account in determining the probability of a particular event. This activity will give students further insight into the gambler’s fallacy.

Mathematics

This activity focuses informally on the concept of independence, asking students to make judgments about whether past occurrences affect future events.

Progression

This activity continues the informal development of probability, drawing on students’ intuitive notions. The idea of independence arises and is connected to the gambler’s fallacy.

Approximate Time

5 minutes for introduction

15 minutes for activity (at home or in class)

15 minutes for discussion

Classroom Organization

Individuals, followed by whole-class discussion

Doing the Activity

This activity follows upon the discussion and defining of independence in The Gambler’s Fallacy. Introduce the activity by mentioning that there are times when an occurrence does affect a future event. You might suggest examples of your own or encourage students to describe some they know.

Read the introduction to this activity as a class. Clarify that students are to determine whether, in each situation, the past will influence the future and to write a paragraph explaining their answer.

Discussing and Debriefing the Activity

Spend a few minutes discussing each of the three situations. In each case, there are reasonable arguments on both sides. Again, the goal of this activity is to clarify the ideas of independence and dependence, rather than to resolve these specific problems.

Here are some plausible arguments that the new situation does not follow the general probabilities.

  • The baseball player’s chances may be greater than 1 in 7 because he is in especially good condition, because he is playing against a team with poor pitchers, or because he is playing in a small park for several games in a row.
  • Your chances of getting a red light may be more than .5 because the lights along Pine Street are not synchronized or because the lights are on a timer and Mr. Bryant always leaves the house at exactly the same time.
  • If the store where you bought the cones is particularly careless in handling merchandise, the chances of finding a broken cone may be more than 1 in 100. It’s also possible that Happy Days exaggerates their quality control.

Review the term independent events, introduced in The Gambler’s Fallacy to describe events whose probabilities do not depend on each other. Help students to see that the questions in this activity can be thought of as asking whether the situations involve independent events.

Key Question

Which of the situations involve independent events?

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