Board Thread:General Discussion/@comment-26006155-20190425222457/@comment-32182236-20190525123258

It's not how likely a hypothesis sounds at first glance. Sure, we start with a base probability, but probabilities change as evidence is discovered.

That's the thing about Bayesion probability.

To find out how evidence changes a probability, we look at two other probabilites. The probability that we'd see the evidence if the hypothesis is true, and the probability that we'd see the evidence period, just by coincidence. We use Bayes' theorem.

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When evidence B is introduced, the new probability of hypothesis A is equal to the prior probability of A, times the probability of finding B, given A.. Divided by the probability of finding B in total.

...That's probably quite hard to understand at first. Perhaps we should test the hypothesis that a coin is rigged? For sake of simplicity, we'll say the coin is infinitely thin, so that it will never land on its edge.

Let's say there's a 1 in 10000 chance that any one coin is actually rigged to only show up heads. We then see a coin flip 12 heads in a row. What are the odds that particular coin is rigged?

The probability of a rigged coin flipping 12 heads in a row is.. 1. So multiplying that with the prior probability gives us 1 in 10000. But now, we must divide!

There's a 1 in 10000 probability the coin is rigged... If it's not rigged, there's a 1 in 4096 chance the coin flips heads 12 times in a row.

1/10000+((9999/10000)*(1/4096)=1/2905.995

Dividing 1/10000 by this number gives us 1/3.441. So there is a 1 in 3.441 chance, or a 29.06% chance, that this coin is rigged. If the coin flipped heads a 13th time, the probability would change to 1 in 2.22, or a 45.03% chance A 14th time, and it becomes 1 in 1.61, pushing us past 50%, up to a 62.1% chance.

But this only applies after we've already gone over the probability of all the alternative explanations explaining the evidence, and it needs to be real evidence, not "this looks cool". After all, the classical four elements are also the four states of matter, the four most abundant materials that we could see on our planet, (though perhaps wood beats out fire), and, if we give air a yellow color, they represent the four hues. These patterns may be what lead alchemy to believe in the four elements. But as we all know... That hypothesis is false. There's well over a hundred, and none of the four elements are actually elements. (Air's a mixture if gases, not just oxygen, so no, air is not an element.)