Intent Students come into class with a limited set of experiences with which to think about probability. In this activity, students gather data to explore, in an informal way, a commonly held misconception about the important ideas of randomness and independent events. In the process, they develop tools to analyze probabilistic situations: namely, simulations and the concept of experimental probability.

Mathematics When you flip a fair coin, you are equally likely to get heads or tails. When you flip a coin repeatedly, heads or tails will come up randomly. If you get three heads in a row, what is the chance that the next flip will be a head? The Gambler’s Fallacy refers to the idea that the next flip is more likely to be a tail because getting four heads in a row is very unlikely. In fact, each flip of the coin is independent. A coin has no memory, and the chance of a head is always the same as the chance of a tail.

Progression This is the second of a number of activities in which students gather data to investigate a probability question. In this activity, students toss a coin many times to investigate what happens after a run of heads or tails. Students pool their data to try to assess predictions that are based on the gambler’s fallacy.

Approximate Time 20 minutes for data collection 30 minutes for discussion

Classroom Organization Pairs to collect data, followed by whole-class discussion

Materials Coins Grid paper Cardstock or index cards

Doing the Activity When students read the introduction to The Gambler’s Fallacy, three points of view generally emerge regarding the roulette strategy of betting on black after a string of reds: A string of reds makes the next spin more likely to be red (as if a trend is happening). A string of reds makes the next spin more likely to be black (as if the string of reds must be compensated for). Previous spins have no effect on the present spin. Take a straw poll to see how many students agree with each of these ideas. Have students read “The Experiment.” Then ask how this experiment is related to the roulette strategy in which gamblers expect a change in result after a string of a given outcome. Have students work in pairs to begin the experiment and record their results. Most pairs will need to take some time to devise a method for counting “sames” and “differents.” Here is one technique you might suggest. Given this series of coin flips:

Students can cut a rectangular “window” from an index card or cardstock and move it systematically over the sequence of outcomes, looking for triplets, and giving results like this (a “same”):

and this (a “different”):

Once pairs have compiled the results of their experiments, total the number of “sames” and “differents” for the entire class.

Discussing and Debriefing the Activity It is unlikely that the number of “sames” will be equal to the number of “differents,” and there is a chance that these numbers will differ enough to be seen by students to support the gambler’s fallacy. With more data, the number of “sames” will get closer to the number of “differents.” The gambler’s fallacy experiments can lead naturally into defining and discussing theoretical probability, observed probability, and independent events. This experiment produces data that students can use to determine the observed (or empirical) probability that the toss after a triplet is the same as the letters in the triplet. The gambler’s fallacy is a fallacy because, according to theory, every flip of a coin has an equal chance of coming up heads as tails, regardless of what has happened before. In other words, each toss of a coin is independent of previous flips. Flipping a fair coin is a random experiment. All of the outcomes of this process—a head or a tail—are equally likely to occur. However, equally likely outcomes do not necessarily result in equal numbers for each outcome when the experiment is done. For example, flipping a coin 100 times is unlikely to result in exactly 50 heads and 50 tails. However, as this experiment is done more and more times, the number of heads will get closer and closer to half the total number of tosses. You can connect the gambler’s fallacy with the central unit problem by asking, How is the gambler’s fallacy related to strategies for the game of Pig? Specifically, ask whether students thought that the chance of rolling 1 became greater if they had already rolled something other than 1 several times. It’s not reasonable to expect to dispel the misconception of the gambler’s fallacy entirely, even if students state unequivocally that they would not be so foolish as to accept it. Even knowledgeable adults operate on the basis of this idea at least occasionally. Sometimes just giving a name to such a mistaken idea can help people identify and avoid it. You might use Fathom surveys technology to pool the data from several classes, click here. [link to TF010204]. With Fathom you can also simulate data for one or many classes [link to TF010205].

Key Questions What do you think about the strategy described in the activity? Why does this activity talk about a roulette wheel but tell us to use coins? Why are the number of “sames” and the number of “differents” so close? Why doesn’t what has happened already affect the next flip of a coin? How is the gambler’s fallacy related to the roulette strategy? How is the gambler’s fallacy related to strategies for the game of Pig?