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# Gamblers Fallacy Simulation

Module by: David Lane. E-mail the author

Begin by answering the questions, even if you have to guess. The first time you answer the questions you will not be told whether you are correct or not.

## General Instructions

The gambler's fallacy involves beliefs about sequences of independent events. By definition, if two events are independent, the occurrence of one event does not affect the occurrence of the second. For example, if a fair coin is flipped twice, the occurrence of a head on the first flip does not affect the outcome of the second flip. What if a coin is flipped five times and comes up heads each time. Is a tail "due" and therefore more likely than not to occur on the next flip. Since the events are independent, the answer is "no." The gambler's fallacy is mistakenly believing that the answer is "yes."

Believers in the fallacy argue correctly that in the long run, the proportion of heads will be 0.50. Or, stated more precisely, that as the number of flips approaches infinity, the proportion of heads approaches 0.50. Doesn't this imply that there must be some natural adjustment occurring, compensating for a string of heads with the later occurrence of more tails? This simulation allows you to explore this question yourself.

You can simulate the flipping of a single coin by clicking the "flip once" button. The number of flips (n), the number of heads, the number of tails, the difference between the number of heads and the number of tails, and the proportion of heads are all recorded and displayed.

To speed things up, you can click the "Flip 100 times" or the "Flip 1000 times" buttons. The "reset" button clears all the data.

The lower portion of the simulation allows you to simulate 25,000 flips at a time. After you click the "Flip 25,000 times and draw graphs" button, the graph on the left will plot the difference between the number of heads and tails as a function of the number of coin flips. The graph on the right will plot the proportion of heads as a function of the number of coin flips.

## Step by Step Instructions

Click the "flip once" button and you will see a simulated coin toss. You will see the results which will show the number of flips (n) which, of course will be 1 after just one flip. Also shown will be the number of heads, the number of tails, the difference between these two quantities, and the proportion of flips that came up heads.

Click the "flip once" button a few times and view the results. Now try clicking the "flip 100" button. You will see the results of 100 coin flips. Take a look at the difference between the number of heads and the number of tails. Is it close to 0? Is the proportion of heads close to 0.5, Click the "flip 1000" button several times noting the result after each time. Is the difference between the number of heads and tails converging on 0? Is the proportion of heads converging on 0.5?

For a more precise answer to these questions, click the "Flip 25,000 times and draw graphs" button. The graph on the left shows the difference between the number of heads and tails on the Y-axis. The number of flips is shown on the X-axis. Does the graph look like it is heading for 0?

The graph on the right shows the proportion of heads as a function of the number of flips. Does it look like it is converging on 0.5?

Click the "Flip 25,000 times and draw graphs" button over and over with these questions in mind. It may take as many as 1,000,000 flips for the proportion of heads to be 0.500 (rounded of to three decimal places) although it will probably reach that value much sooner. Does the difference between the number of heads and tails converge on 0 the way the proportion of heads converges on 0.5?

This simulation shows that the difference between the number of heads and the number of tails does not converge on 0. There is no "compensatory force" that makes up for too many heads (or tails) by causing more of tails to occur. The reason that the proportion of heads approaches 0 is that the difference between the number of heads and tails becomes very small compared to the total number of coin flips.

## Summary

When an interval scale such as sugar content is mapped onto a rating scale such as judgment of sweetness, the resulting rating scale is probably not an interval scale. For most real-world situations, the means of ordinal-level rating scales allow valid conclusions about the direction of the means on the interval scale. However, it is theoretically possible for means on the ordinal scale to be in the opposite direction from means on the interval scale. Experts disagree on the importance of this in real-world data analysis. We believe that the chances of misintrepretation with real data are extremely low, and that it is only with contrived artificial data and mappings of the interval to the ordinal scale that these problems occur.

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