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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Possibilities vs. Realities. Adventures With 3 Coin Flips: Part 3

Flipping a coin three times seems like a simple process.  But there are myriad complications that can arise.  In Part 1 of this series, we saw that data sampling for coin flips can influence how results are interpreted.  In this post, we will look closely at how probability assessments (possibilities) can lead to propositions deviating from the reality ensuing when coin flips are actually carried out.  We will consider the concept of ‘alternate universes’.  Who would have guessed that flipping a coin three times would go there?

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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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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.

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The Growth of Hierarchy and the Death of the Free Market

Economics from the Top Down

Do you believe in free markets? Do you think that unfettered competition is the best way to organize society? If so, this post is intended to shake your faith. No, I’m not going to argue that free markets are bad. Instead, I’m going to show you some evidence that may disturb you.

I’m going to show you that as societies develop, they kill off the free market and replace it with a command economy. Let me be clear about what I mean. To most people, a ‘command economy’ means government command. But I use the phrase to mean business command.

A business is an island of central planning — a command economy living in the free-market sea. Yes, businesses sell things on the market. But inside the business, hierarchy is the organizing principle. Employees in a large firm do not barter and trade with each other. Instead, they obey a…

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The Republican “betrayal”

The truth about money

Republicans only demand austerity and a balanced federal budget (and only indulge in their “debt ceiling” antics) when a Democrat is in the White House, or when Democrats control Congress.

For example, austerity mania raged in Washington from late 2010 to 2012, because a Democrat (Obama) was in the White House. Perhaps you remember Obama’s “cat-food commission,” and “grand bargain,” plus all the chatter about “sequesters.”

Before that, there were the disastrous four years of Bill Clinton’s budget surplus (1998, 1999, 2000, and 2001).

Republicans talk about austerity, but only Democrats put it into action.

When a Republican is in the White House, “deficits don’t matter,” as Republican Vice President Dick Cheney told Treasury Secretary Paul O’Neill.

Indeed, on 31 Dec 2002 President Bush fired Secretary O’Neill for opposing Bush’s spending spree for the invasion of Afghanistan and (in March 2003) Iraq.

This pattern never changes. Therefore it amuses me to see “fiscal conservatives” whining about Republican “betrayals.”

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Your House Can Pay You in Retirement

After years of making mortgage payments some retirees are nor getting payments back from their homes.  Reverse mortgages have been a possible part of retirement planning for many years.  They are a way for retirees to use equity in their home as a source of income payments.  The plans have often functioned in the form of an annuity with fixed monthly payments for the life of the reverse mortgage agreement or with a fixed sum payout at the closing.  Various options for the structure of reverse mortgages – some are discussed below.  There are some, however, for whom the reverse mortage option is simply not viable.  The reason for that has to do with the recent housing bubble and the ensuing collapse.

How Your Tax Dollars are Spent

Accounting Degree Online has created an informative infographic about U.S. personal taxes.

Click “Read the rest…” to see the entire presentation.

Tricky Social Security Questions Answered

This is a Special Report by Elaine Floyd at horsesmouth.

Topics:
Suspending benefits before full retirement age
Survivor benefits on a first marriage
Going back to work after starting Social Security
How will the recession effect Social Security benefits?

College Loan Debt is a Big Problem for Borrowers, Lenders and Government

MSN Money and The Wall Street Journal have combined to produce an article detailing some of the problems that exist for those with college loans debt. Among the situations cited is a 41-year old MD with $550,000 outstanding college debt. The debt was much less (about $250,000) when this individual graduated from medical school in 2003, but has ballooned to the present amount through mismanagement.

Another case mentioned is a laid-off factory worker with $120 a week garnished from her $300 a week unemployment check to apply against her son’s college loan debt. The son is also unemployed, having lost his $29,000 a year job 8 months ago.

A third case describes a college loan debt that has grown from $28,000 to more than $90,000, with monthly payments that were originally $230 now $816.

How do such cases happen?