Bayes Formula - and why it's important

[TheWorstPlayer]

OK, I'm not great poster like Fim so I'm not working up a whole worksheet (that post was awesome, by the way) but I can if there is sufficient demand. Here is the quick and dirty version though.

Let's say that there are two mutually exclusive, completely exhaustive events: good weather, bad weather. Each happens 50% of the time.

Now let's say that there are two, and only two, possible actions: go out, stay in.

Now let's say that when it's bad weather, we go out 20% and stay in 80%. Similarly, when it's good weather, we go out 80% and stay in 20%.

Now let's say that I told you that yesterday, I went out. What is the probability that is was good weather yesterday?

Well, if we had no information about my actions, then the probability would be 50%. That is technically called the a priori probability. However, we DO have information about my actions and we can use that information to improve our estimation of the probability of good weather. Our new estimate, which takes into account the additional information that we have in the form of my action and the impact of the weather on my actions is called a posteriori probability.

Bayes Formula (which I could easily derive, if people are really interested) states that:
The probability of A given B is equal to the probability of B given A times the probability of A divided by the probability of B given A times the probability of A plus the probability of B given not A times the probability of not A.

So if we substitute 'good weather' for A and 'go out' for B we have what we want to know:
The probability of good weather yesterday given that I went out is equal to the probability of going out given there is good weather times the probability of good weather divided by the probability that I go out given there is good weather times the probability of there being good weather plus the probability that I go out given bad weather times the probability of bad weather.

So, in notation,

P(good|out)=[P(out|good)*p(good)]/[P(out|good)*P(good)+P(out|bad)*P(bad)]
P(good|out)=[80% * 50%]/[80% * 50% + 20% * 50%]
P(good|out)=40%/50%=80%.

So, what does this tell us? This tells us that if we didn't know whether or not I went out, we would say that it had a 50% chance of being good weather yesterday. But once we know that I went out, and we know how the weather effects the probability of my going out, we can say that since I went out yesterday it had an 80% chance of being good weather yesterday. This result that it is more likely to have been good weather since I am more likely to have gone out in good weather should be intuitive to everyone.

How does this relate to poker? Well let's say that you raise and someone pushes over the top. You know that they will do this with AA or KK but no other hands. So is it equally likely that they have AA and KK since there are equal combinations? Maybe but maybe not. Let's say that they will push over the top 100% of the time with AA but they will only do so 80% of the time with KK. You tell me how often they will have KK here.

[amoeba]

I think there is a fallacy here.

if he pushes 100% with AA and 80% with KK then its not equally likely that he has AA or KK.

in other words we have to throw out the initial condition.

[swolfe]

44.4_%

good post

[passion]

(4/9)'ths

[TheWorstPlayer]

This is my point. I never said they were equally likely.

[TheWorstPlayer]

[ QUOTE ]
good post

[/ QUOTE ]
thanks. i hope it's helpful. i thought this was so fundamental that everyone on here knew it, but apparently i was wrong. I'm sure everyone does it instinctively, but it is good to have some formal structure to your thinking, IMO.

[amoeba]

ok just making sure.

they are only likely interms of likelyhood to happen rather than for this particular player.

[passion]

Ok - I answered the question, but I have not been convinced that Bayes' Theorem is particularly useful for poker. The problem is that for typical applications it is very difficult to know the prior probabilies need to calculate the posterior probabilities. In your particular expample you knew the player pushed AA 100% of the time and KK 80% of the time, but in reality this is more than you ever know. At best you can do obtain an ad hoc estimate of these probabilities which is otherwise known as making a read

Passion

[TheWorstPlayer]

[ QUOTE ]
you can do an ad hoc estimate of these probabilities which is otherwise known as making a read

[/ QUOTE ]

[Mackerel]

Very nicely explained.