Let’s assume that I think all video poker machines in Nevada are fair — and when I say fair I mean that each unseen card has an equal probability of showing up next. Now let’s assume that I play 200,000 hands of single line Jacks or Better and do not hit a royal. That’s about five cycles. The chances that I will go the next five cycles without a royal on a fair machine are about 1-in-150.
You then ask me if I still think the machines are fair. My answer is going to be, “Yes, the machines are most likely fair.” My ‘prior belief’ dominates my belief system.
Now let’s look at Joe. Joe is very uncertain about whether or not the machines in Nevada are fair. It’s 50-50 as far as he’s concerned. Now he goes through the same 200,000-hand dry spell. Ask Joe whether or not he now thinks the machines are fair and he’s basically convinced that they are not. His prior belief gave him no conviction whatsoever of the machines’ fairness. The current evidence of the machines’ unfairness seems pretty overwhelming to him.
The same set of facts led Joe and me to very different conclusions.
We see this all the time in political discussions. Before the last election, if you took a strong supporter of Romney (i.e. one whose prior belief said that the country would be better off if Romney were elected rather than Obama) and let him discuss politics with an Obama supporter (i.e. one whose prior belief said the opposite), they would agree on almost nothing. It’s possible that both of these supporters were thinking logically given their prior beliefs, but their prior beliefs are so different that no common ground is possible.
One of the foundations in the field of Probability and Statistics is something called “Bayes’ Theorem.” This theorem calculates the probability of something GIVEN your prior beliefs. The theorem itself is not particularly complicated, but it is heavier math than I want to go through today. Another term for Bayes’ Theorem is ‘conditional probability.’
You sometimes hear the phrase, “the facts speak for themselves.” This is usually not true. How someone interprets a given set of facts depends on what his prior beliefs are. Gaming attorney Bob Nersesian was recently a guest on my radio show (Gambling with an Edge — details and old shows archived here) and was talking about how prosecutors sometimes reach conclusions that are very different from those reached by defense attorneys. The prosecutor often seems to believe that if an advantage player is arrested, he is most likely guilty, while the defense attorney believes that if such a player is arrested, the casino was likely guilty of over-stepping their legal authority. Same facts. Very different conclusions.
Nersesian continues to be flabbergasted at how narrow-minded prosecutors can be. There’s no reason to be so amazed. It’s Bayes’ Theorem at work demonstrating the strength of prior beliefs.
I sometimes find that Bayes’ Theorem affects how readers take my writings. There are some players who believe that I’ve made a large contribution to their video poker knowledge and that if I recommend something, they are inclined to follow that recommendation. There are other players who believe that I’m (pick one or more: arrogant, selfish, over-rated, two-timing, money-grubbing, etc.) and there is nothing I can recommend that they are going to follow. That’s Bayes’ Theorem again. Different prior beliefs. Different final conclusions.
Almost all of us regularly use Bayes’ Theorem in our personal lives. You’re on the road and you’re hungry and you come to a Subway sandwich shop. You’ve never been in this particular Subway before, but you likely have a strong prior opinion about what you’ll find there. Some people’s prior belief will tell them: “It’s healthier than other fast food places.” Others’ prior belief will say, “No fast food place is good enough to me.” Yet others’ prior belief will say, “It doesn’t matter. I’m hungry. Any place is okay.” That prior belief will go a long way toward deciding where you are going to eat today.
Or you come to an off-brand gasoline station that has gas priced 4¢ a gallon cheaper than the Chevron station across the street. Some people have a prior belief that says, “All gasoline is the same,” and so the price differential is the determining factor in the decision. Others have a prior that says, “I trust Chevron. I don’t trust the no-name gas.” For these people, the price difference is not a factor in their decision.
This is a VERY useful way to predict how good we think something will be. We each make hundreds of decisions a day (or more), and thinking through every one of them from scratch each time will severely limit how much we can get done.
However, in addition to being useful, Bayes’ Theorem can also lead to bias and prejudice. If I see a couple of guys who look like they might belong to the Hell’s Angels motorcycle gang, I’m going to give them a wide berth. I have a prior belief that says these guys could hurt me badly if they become irritated with me and I don’t want any chance of that happening. On the other hand, they could both be very nice, friendly men who, for whatever reason, appear to me to be gang members. I might not discover their true nature since I would avoid them given my prior beliefs.
Many people have prior beliefs related to skin color, religion, sexual preference, age, size and shape of certain body parts, nationality, etc. Many of us try to treat everyone equally — but it’s very difficult to overcome our own prejudices.
Bayes’ Theorem doesn’t give any guidance on how to avoid our biases. It only helps us predict things given our biases. So it is very useful. But in some contexts, very limited.
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