Fred Hoyle, an accomplished astronomer, radio personality and science fiction author, was most famous for being wrong. He never accepted the current cosmology, coining the name Big Bang in an attempt to deride it. And to the delight of creationists, he also never accepted the role of chance in the evolution of life.

What are the odds, Hoyle asked in his 1983 book Intelligent Universe, that a whirlwind passing through a junkyard would happen to assemble a 747? Hoyle claims they're comparable to the odds that life evolved by chance, a modern variation on the argument from design, the last of Saint Thomas Aquinas' five proofs of the existence of God.

Another argument in the same vein makes an analogy with Rubik's cube, for which it's at least possible to compute the probabilities. In an article in the November 19, 1981 New Scientist, Hoyle writes,

Now imagine 10⁵⁰ [1 followed by 50 zeroes] blind persons each with a scrambled Rubik cube, and try to conceive of the chance of them all simultaneously arriving at the solved form. You then have the chance of arriving by random shuffling of just one of the many biopolymers on which life depends.

The most eloquent refutation of this statistical argument is given by Richard Dawkins, and when people ask me about it, that's where I refer them. But some are too impatient to read the careful exposition in Dawkins' books, so to give the reader the flavor of his arguments, I'll turn to another amusement, Milton Bradley's dice game Yahtzee.

In Yahtzee, players throw 5 dice from a cup and get a score based on what comes up.

The odds of throwing 5 sixes is 6^-5, or 0.0001286. You'd have to throw the dice 3888 times to have a 50% chance of getting 5 sixes. So 5 sixes in one throw is highly improbable.

But in Yahtzee, you get more than one throw per turn, and you get to leave on the table the dice you like, and only throw the rest. Now we have to ask how many throws it takes, on average, to accumulate 5 sixes. This is more difficult to calculate (and I'll show how I did it in a minute), but it turns out you need, not 3888 throws, but only 12.

Think of the sixes as the organisms that survive, and each throw as the next generation of dice.

To borrow a metaphor from Dawkins, you don't get to the top of a mountain in a single step. You get to the extremely improbable top by taking lots of small, much less improbable steps.

The Calculation

You shouldn't have to take my word for the 12 throws, so I'll walk through part of the calculation.

On the first throw, you have 5 dice. The odds you'll throw at least 1 six are the same as one minus the odds that all of the dice are not six, or

1 - ( 5 / 6 )⁵ = 0.598,

better than 50-50.

The probabilities are calculated as

p = C( 5, n )  ( 1 / 6 )ⁿ  ( 5 / 6 )^{( 5 - n )}

where n is the number of sixes and C( i, j ) is the number of possible combinations of i things taken j at a time,

C( i, j ) = i ! / j ! ( i - j ) !

The exclamation mark is the factorial,

i ! = i ( i - 1 ) ( i - 2 ) ...  (3) (2) (1)
5! = 5 x 4 x 3 x 2 x 1 = 120

For each of the 6 branches, you build another table with all of the possible outcomes after the second throw. And then you branch again.

This is pretty tedious, though. You can take a short cut on the computer by creating a Monte Carlo simulation. Tell the computer to play the game a large number of times (say 50,000), and keep track of the number of throws it takes each time. That's what I did.