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Bayes Theorem

Bayes TheoremThere are many sports handicapping experts out there, or so they want us to believe.  How do we know these guys are who they say they are? It’s hard, if not impossible, to track down reliable records of most handicappers. The industry doesn’t have a central organizing body that provides a license of documents picks. However, even when we can’t get our hands on the actual records we can use a nice piece of statistical analysis to get a clearer picture.

Bayes Theorem is a statistical technique that allows us to see the probabilities behind the results.  While taking a finite math class in college I quickly learned health examples are the leaders when it comes to teaching Bayes Theorem.  Imagine this:  A man in Country ABC sees his doctor to determine whether he is suffering from disease Z.  He takes a test, and comes back positive for disease Z.  The doctor who gave this test went in knowing that 5% of people from country ABC have disease Z.  We also know that when someone has the disease the test will show a positive result 100% of the time.  Yet, if the person does now have disease Z a positive result will still come up 10% of the time.  These numbers are particularly important, especially the false positive.

At this point most people would be pretty heart-broken, they just received the test results and they came back positive for disease Z.  If he has the disease there is a 100% chance it will be positive, and if he doesn’t have it there is only a 10% chance of error.  So his positive result means there is approximately a 90% chance he has the disease, right?  Not exactly.  This is the ‘common logic’ approach  to our question, but it doesn’t stand up to a mathematical test.  When we apply Bayes theorem to this problem we see a different answer all together.

Here’s the stat that surprises everyone; only about 1/3 of the positive test results actually have the disease. There is 5% of the population that has the disease, so if 10,000 people are tested, we know beforehand that 500 will test positive for disease Z.  Yet, 10% of the remaining 9500 will test as a false positive.  This means that out of the remaining 9500, 950 will test positive and not have the disease. So the total number of positive test results from our sample size of 10,000 is 950 + 500 for a total of 1450 positive test results.  500 / 1450 = 34.5%.  So only 34.5% of the positive test results actually have the disease.  Our man from Country ABC need not hang his head!  He still has a very good chance of not having the dreaded disease Z.  The general formula is:

P(A|B) = [ P(B|A) P(A) ] / P(B)

This reads the probability of A given B = the probability of B given A multiplied by the probability of A, this number is then divided by the probability of B.

So how do we apply this to sports handicapping and sports handicappers?  Well, first we know that with so many handicappers out there, there are always going to be bums who have gone on a recent hot streak that makes them look appealing.  Let’s say a guy picks 100 NFL games for the year and wins 57 of them.  57% is really good, maybe you should buy his picks.  But look at it this way, would you trust the batting average of a guy who only had 100 at bats, or the completion percentage of a quarterback with only 100 attempted passes?  Of course not, that number is likely way off from what his “true” abilities are.  It is only through a larger sample size that you can see this.  Let’s use some actual numbers:  Suppose that 5% of sports bettors can actually win against the spread in the long run, and 80% of those are winners over a 2 year period.  95% of bettors are pretenders who are just flipping quarters, and 20% of them are winners over a 2 year period (even a blind squirrel finds a nut once in a while), then this means that 82.6% of bettors who are winners over a 2 year period are actually no-talent bums who are long term losers!

Sports handicapping is a tough business, most don’t have what it takes.  Don’t fall for the sales jobs, guys raving about their “lock of the week” or their “play of the day”. Think about it, these guys call you up and say things like, “as much as you want to win should be how much you bet on this game.”  I bet a lot of them would have installed Michigan as a “lock” against Appalachian State a few years back too.  I thought Michigan would have won too, but we don’t use words like “lock” because we realize anything can happen.  It’s my Big Ten Play of the Year…sure it is.  The difference is we have identified what is most likely to happen and found the correct play to exploit it.


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