Brain-Twister: Tossing A Coin - Not So Simple...
Note: This article is secretly a post about Management, and how people think. But I won't tell you that. You see, people take Management very seriously, and they shut down at the mere sight or sound of the term. But if we re-frame the same content as a Brain-Twister, everybody wants to play! Don't speak a word of this to anyone!
When I was a mere tadpole in the New York System (which system has subsequently become a tragic) study in entropy, I was taught about probability. We were all indocrinated about the difference between "independent events" and "dependent events."
Every one of us was absolutely clear that each toss of a "fair coin" (balanced, and not weighted or otherwise structurally altered) was an independent event, with the probability of the coin's landing on Heads being equal to .50 (fifty percent) and the probability of the coin's landing on Tails also being equal to .50 (fifty percent). Each coin toss was independent of any other coin toss, and the probability of a tossed coin landing with the heads side up was exactly the same as was the probability of the same tossed coin landing with the tails side up.
It was an immutable, irrefutable proposition.
What they never told us was that these probabilities were "statistically determined" based upon a "very large" number of tosses. As we all know, occasionally you can toss a coin several times and get just heads, just tails, or some combination that was not quite 50/50. Intuitively, we believed that given enough tosses, the number of times that heads would come up would always be equal to number of times that tails would come up. The notion of an infinite number of tosses tending toward the sacred 50/50 ratio was never questioned -- yet, when tossing a coin just a few times, the outcome was, in fact, very seldom, 50/50.
Here's my question, framed in several different ways:
If you were betting on a coin toss game (which was unfixed, honest and legal), and the first two tosses turned out to be heads, would you bet on heads for the next toss? How about if the first four tosses turned out to be heads? How about if the first seven tosses turned out to be heads (statistically unlikely, but definitely possible)?
Here's my observation:
As the number of heads in a row increased, the odds of your betting on the next toss coming up heads would be reduced. In your mind, you are probably thinking something like this..."Since the statistic is eventually supposed to average out to 50/50 the REAL LIFE odds of the next toss resulting in landing on heads is decreasing with every toss. The next one's JUST GOTTA turn up to be tails."
What is the reason for this type of thinking? It is because we feel that the individual tosses are not truly independent, and that the earlier tosses are "statistically forcing" the outcome of the next toss.
If you had never been taught about this 50/50 statistical tendency (let's say that you were some cave-dwelling type of individual, as were many of my relatives, and that you only knew what you observed) -- you would then behave in just the opposite manner. With each toss resulting in heads, you would become increasingly convinced that the next toss would ALSO be heads. Your reasoning would be based purely upon your observation of a relatively small number of occurences.
Here's My Confession:
After seeing an increasing number of heads coming up in a row, I would become increasingly likely to bet on tails...as if the tosses had to support the statistic based upon the infinite example. I believe that somehow the universe will force the next toss to bring my observations into line with what I'd been taught. This means that I do not fully believe that each toss is truly independent of the next.
Here's Some Food For Thought:
Both the cave-dweller (my great-uncle Farkas) and I would be biased based upon either pre-conceived notions or inadequate testing. We would both be using logic, but we would both be wrong.
Here's a Bit More:
People make decisions either based upon over-education or under-experience.
As always, I welcome your comments.
p.s. Please feel to express your thoughts regarding this post in the Comment Section which follows in the itty-bitty hyperlinks under this article.
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