Conditional Probability

I was reading an article recently about the enormous Powerball Jackpot which said that you are 25 times more likely to win an Academy Award(25 million to 1) then to win the Powerball Jackpot (175 million to 1).  I dislike it when writers use statistics that are right factually, but that are wrong when applied in the right context.  I am going to explain why the writer was wrong, and how the right thinking can have a big impact on the way you make decisions.

First, what are the true odds that I will win an Academy Award?  Realistically, about zero.  I am not an actor, do not have plans to become one and am in no way involved in the production of movies or films.  On the other hand, what are the odds that Anne Hathaway will win one?  Pretty damn high (my wife thinks that she will win this year for Les Mis).  So, putting the odds of both myself and Anne Hathaway at 25 million to 1 is seems to be wrong.

So, how do you go about figuring out what the right answer truly is?  The answer is Bayes’ Theorem and it has applicability to many things, including drug testing, medical experiments and our everyday decisions.  Basically, you include the specific data you have about a situation and compare its probability to the general data you have.  The formula is a bit complicated and there are many online calculators available, but it is best explained by the example here.

How does this help us figure out the true answer to the Academy Award question?  Well, instead of figuring out the odds that anyone will win an Academy Award and applying it to me, you take the information you know about me(that I am not an actor) and apply it.  What are the odds that I will win an Oscar, given that I am not an actor.  Miniscule.  What are the odds that Anne Hathaway, given that she is an A list actress with a major film role upcoming will win an Oscar?  Pretty high!

There is a slightly more complicated example of Bayes’ Theorem that applies to medical testing that is very important since its impact can be outsized in its impact on our lives.  You read that there is a rare cancer that occurs in .1% of the population and that there is a test that will let you know if you have it that is 99% accurate for people with the disease.   1% people who do not have it will also get a positive result.  Should you take the test, and if you come back positive, what is the likelihood that you have it?  Most people would say they should take the test(I am not going to weigh in on that) and that they are 99% likely to have it, but the number is much, much, lower than that and should be considered before deciding whether or not to take the test.  Here is the math behind it:

There are 350 million people in the USA and if everyone were tested, 350,000 people would be expected to have it(.1% of 350 million).  Of those, 99% would come up positive, so 346,500 people of the 350,00 who have it would come up positive on the test.  Of the 350 million people tested, 3.5 million will test positive for the cancer when they are really negative.  So, to figure out the odds of actually having the cancer after testing positive for it  you would take the number of correct positives and divide it by the total number of positives or 346,500/350,346,500 or ~1 in 1,000!

This uses the same Bayes’ formula as the Academy Award example (the given in this case is that you have a positive result) and shows that a test that seems great on the surface(99% accurate for the disease!) actually leaves 99.9% of the people who test positive thinking they have the disease when they do not.

This blog is getting long, so just a quick note on how Bayes’ Theorem applies to poker.  Sometimes I have to make decisions without as much information about my opponents as I would like.  I learn as much as I can about them as quickly as I can, but sometimes, a situation comes up that requires me to make an important decision within the first few hands of having played with someone.  So what do I do?  Well, instead of pretending I have no information, I use the information I do have (limited as it may be) and apply Bayes’ Theorem.  For example, when someone sits down and raises his first 3 hands I now know that it is way more likely that he is a maniacal player than if he sat down and folded the first 3 hands.   Knowing how rare it is that a “normal player” would raise 3 hands in a row as compared to how often a maniacal player would, helps me allow for the data I have and lets me change my actions accordingly.

If you have any questions please email me at Jason@PokerConsultant.org.

Bonus question: Do the odds of Anne Hathaway winning the Oscar change when you know that my wife, who has won several Oscar pools, and is very good at picking the winners, thinks that she will?

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One Response to Conditional Probability

  1. wife says:

    you have a wife??

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