### About Bayes

A quick note for B on why Bayes' theorem is useful. Not going to talk about what the theorem actually is much or go deeply into the maths. Just a backgrounder so you get it.Let's say there's a disease that 1% of the population has, and a test for that disease that is 90% accurate.

John is tested positive for the disease. Bad for John, right?

Well, think about this. Let's say we tested 10,000 people. We know that 100 of them have the disease and 9,900 do not.

The test will be positive for 90 of the people that have the disease and for 990 of those who do not. (If the test is 90% accurate, it will give a positive result for 9/10 of those who have the disease, but will also give a positive for 1/10 of those who don't.)

So of the positive results, 990/1,080 are false positives. That's 91%!

So John is only 9% likely to have the disease.

When I came to Australia, I had a test for HIV. Now let's say the figures are just as they are here: 1% infected, 90% accuracy in the test. Let's look at our 10,000 people again. Of the 100 that have the disease, 10 will return a false negative and of the 9,900 who do not, 8,910 will return a negative. So it's very likely I do not have HIV: 10/8920 = 1/10th of a percent.

Why do I think this applies to poker? Well, let's say that after ten hands, we have a guy who has raised three times. Let's simplify the world of poker to two types of players: tight and loose. The tight guys raise 10% of their hands and the loose guys raise 30% of their hands (the real poker world is not like this but we do have a priori knowledge of the types of players we are likely to encounter). It should be clear that the latter guys are 3x more likely to raise on any given hand. It's like this: if each player is dealt A7s, the first guy just folds, the second raises it (it's outside 10% but inside 30%). Each hand has an equal likelihood of being dealt.

I won't do the maths, but we can work out how likely it is that any given raise is from a loose player given our a priori knowledge of the game (if the players are split 50/50, it's 75% obviously), and we can work out how likely it is that over 10 hands the tight player will have picked up three hands from within the 10% (my learning of probability does not stretch to doing this but it's an easy bit of maths that I just didn't absorb). I actually posted this example previously but I can't remember what the post was. The outcome is that a player whose stats are 30/30 over 10 hands is much more likely to be a loose player than a tight player (in fact, very few players are as loose as 30/30 and the result is like the negative in the test, not the positive).