Scenarios Affect Probability

Let's try to calculate the probability of some event. To keep it dead simple, we'll use coin flips. But we'll calculate the answer in two different ways, and then I will point out how these ways relate to Evolution.

The Problem:

How long will it take to flip 166 coins, until all of them come up heads?

[Or, to say the same thing the other way up: If I flipped for a day, what would be the chance I would succeed in getting 166 coins to come up all heads?]

Using Scenario One:

Put the coins in a neat line, and go down the line. Give each one a nice thumb-snap flip, catch it and put it back down. When you get to the end of the line, if they are all heads, stop. Otherwise, start over.

I tried this for a little while. Each coin took two or three seconds, so 166 coins would take 300 to 500 seconds - let's say, 6 minutes. The chance of the row being all heads is one chance in 2166, which is almost exactly the famous one chance in 1050. So, flipping coins 24 hours a day, my handy computer says I am likely to be finished in a billion billion billion billion billion years.

In one day of trying, the probability of my getting them all heads, is effectively Zero.

Using Scenario Two:

Take 166 coins in your hands, or in cup or pie plate. Throw them against the back of a couch, so that they bounce, and all land on the seat of the couch. Remove the ones that came up heads. Scoop up the rest, and repeat. Stop when there aren't any to scoop up.

I tried this with dimes. The slow part was grabbing the coins that came up heads: that took a second or two, each. Each of the 166 coins only had to be grabbed once, so I calculate the whole process should take a few hundred seconds. And that's about how long it took. I did it several times, and I was always done in less than four minutes.

In one day of trying, the probability of my getting them all heads, is effectively One (dead certainty).


Apparently Scenario Two is about 1050 times faster than Scenario One. I would like to make three points here.

The first point is that scenarios are more slippery than beginners think. When you saw my problem statement, above, did many scenarios leap into your mind? Or just one?

The second point is that calculating the probability of the wrong scenario can give a very wrong answer. A wrong scenario may not be "good enough".

The third point is how all this relates to Natural Selection. Beginners often assume that evolution is supposed to work just like Scenario One - nothing happens, nothing happens, and then one day, Bang, there is a mutation which turns a dog into a cat, or some such. But Scenario Two, which collects and accumulates many little bits of good luck, is much closer to the actual scientific theory. Have a look, for instance, at a "weasel" program which gets a 1045-fold speedup from using cumulative selection.

Last modified: 23 July 2013

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