Learning to Count at an Advanced Age. Adventures With 3 Coin Flips: Part 4.

The process of understanding the complexity of many 3-flip sequences for tossing a coin is about to become more difficult.  This is because are going to enter the arcane world of permutations and combinations.  As a result, this is counting on a scale far more complex than that learned in elementary school.

Counting Results for Multiple Coin Flips

As we have seen previously, there are 8 equally probable possible results for the 3-flip sequence.  The general formula for the number of possible results for a series of events when each event has the same number of possible outcomes is n^r.  In this, n is the number of possible results for each event, and r is the number of events in the series.  Thus, for a 3-times repetition of an event with a binary result, the number of possible results is 2³. The attractiveness of using a 3-flip sequence rather than 4, 5, 6, or more is the rapid increase in possible results to be considered. Each flip added doubles the number of results possible.  By the time one gets to 6 flips, the number of possible results is 64.

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Connecting the Micro to the Macro. Adventures With 3 Coin Flips, Part 2

In Part 1, we saw that increasing the observation window changes the results for the occurrence of tails following heads.  That raises the question: How does the micro (small observation window) relate to the macro (large observation window)?  More specifically, what is the relationship between results from a small observation window (three flips) and those from a large observation window (100 flips)?

Many Collected Micro Events Are NOT The Same As A Macro Event

Blair Fix (Is Human Probability Intuition Actually ‘Biased’?) shows that the apparent bias in favor of tails following heads decreases as the observation window increases.  Fix goes from an observation window of 3 flips to 5 flips, 10 flips, and ultimately 100 flips.  When Fix repeats 3-flip sequences many times and accumulates the results, these accumulations are unconnected to the results for any other n-flip sequence(s).  The accumulated micro-events of one type (type = number of flips) are unrelated directly to any macro-event constituted by many successive coin flips of other types.

Let’s recall two graphics from Part 1 that show the results of simulated iterated coin flips using random number generation.

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