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.

Read the original post.

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.

Read the original article.

The Gamblers’ Paradox – Adventures With 3 Coin Flips, Part 1.

Many gambling activities involve betting on events for which the outcomes obey rigid, specified odds.  When there is no mechanical bias, roulette wheels have fixed odds, including some that are binary (50:50), such as red vs. black. Betting on the flip of a coin is likewise a binary 50:50 proposition:  heads or tails.  Why is it, then, that there is a propensity for some gamblers to place wagers in a pattern conflicting with the known 50:50 odds?  For example, after a string of blacks on a roulette wheel, why do some gamblers keep increasing the amounts bet on red with each succeeding black?

Such behavior is related to the concept that if some process deviates from a known probability for a period of time, future events will counter that deviation in what is called “reversion to the mean”.  However, the expectation that such a process will start with the next process event cannot have a probability different from the known odds.  If someone flips 10 heads in a row, the next coin flip still has even odds of heads or tails.

Australian economist Jason Collins has considered a coin-flipping problem involving 3 flips per sequence. He shows that by choosing the sample size that he has here, a bias is introduced such that the probability of a head being followed by another head is only 42%, not 50% expected for random coin flips.

Read more of the original post.