May 25, 2010
Humans Are Naturally Poor at Gauging Probability
Now that those issues are out of the way, I'm going to bring up a topic that has long been on my mind, but I have yet to really talk about it. The idea for this post came to me one day when I was playing crib (a card came), with my aunt. We were dealt a hand where I got three 5's and she also got a 5 (There being 6 cards in each hand). She thought that this was miraculous and complained that the deck wasn't shuffled properly, or that something that shouldn't have happened did indeed happen. There is a mistake in this belief that, if not already apparent, will become apparent quite soon, but first I will talk about three examples I know that are semi-akin to this way of thinking, and I think it shows how people are naturally poor at judging probability.
The first one comes from a great webshow, which I will unabashedly endorse, called Scam School with Brain Brushwood. I forget which episode it comes from (I found it on youtube), but it is a scam that involves my favorite prop, a deck of cards. The other person can shuffle the deck and you just bet them for a drink that after they are done shuffling that two of the same cards will be side by side. There is no trick to this, it is just that the odds are that two of the same card will be beside each other.
Well here is an example of my research leading me to find out my beliefs were wrong. The odds of the same card being beside each other seems to be about 48%, which while higher then I suspect people would think it would be, is actually a little too low to be betting for, although it wouldn't be a bad bet at a casino. Anyone can take my word for it or read this and find out what I did (I also verified it from other sources). So Brain was wrong here, but I still like his show.
Anyway that leads me two my second, and hopefully more correct example of the coin flip example. This one is an interesting example where a teacher asks his math students to either do their homework or fake it. The homework is to flip a coin 200 times and record heads or tails for those flips. At a glance the professor can tell whether the student faked the flips or really did their work. The reason the professor can tell is that people are poor at judging probability. In a trial of 200 coin flips there is what is described as an overwhelming chance that there will be a run of six of the same outcome. Most people, including the students that faked their tests, would think that 6 heads or tails in a row would be unlikely and created results that reflected that. Even the math students were poor a judging probability. If you want to read more about it and learn about something close to this idea named Benford's Law just click here.
My third example is the one that you are most likely to have heard previously because recently it has been making its rounds around the internet. It is what is referred to as the Monty Hall Problem, named from the Game show Lets Make a Deal, which had the host Monty Hall. The set up is there are 3 doors to choose from and you have to pick a door. After you have picked a door you have the option of keeping the door you picked or switching to the only other unopened door. This is where peoples horrible appreciation for probability comes in. Now it seems like it is a 50/50 proposition because there is one prize and two doors, but in reality, but taking a door away and offering you a door he is giving you two doors for the price of one. The reason is that there is only one prize, and he won't revile a door where the main prize is, so you are getting the chance of the two doors combined, even though the one door has been show to have a fake prize. Switching gives you a 2/3 chance and sticking with your first choice leaves you at the 1/3 you had before a door was revealed. I know this sounds counter intuitive, and people have had many problems with this, but wikipedia does a good job in explaining why that is if you didn't understand my explanation.
What these three examples show is how easily people error in judging the odds for something, and I don't think I'm in anyway different. I had to look up the information about 2 of the same cards (any pair) being beside each other, and even Brain Brushwood, who was putting his wallet where his mouth was, was getting the odds wrong. It just goes to show why the lottery and casino's can make so much money, people have a really hard time calculating the odds of something, and that is when they are unclouded of beliefs that would lead them to think that they are 'lucky' or that they are 'due'.
Anyone should be able to see the problem in my Aunts logic now. It is the same mistake that people were making when they were faking their coin flips, they don't judge the probability correctly and think that anything like 6 heads in a row or all the 5's being dealt out is something that shouldn't happen, no matter how many flips are done or hands are dealt out. This is the belief that something 1 in a million should never happen, even if that thing is done a million times.
I'm sure people have some interesting stories about people misreading the odds so feel free to post them in the comments, thanks for reading.
- The Moral Skeptic