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This time I'd like to talk about the Gambler's
Fallacy. In researching this topic, I discovered that just about
anything gamblers believe that is contrary to mathematical reality
is considered a gambler's fallacy. Let's look at a few of those
fallacies over the next few weeks.
The
Odds Have to Even Out
This
is the most common gambler's fallacy. The mathematicians call it
The Doctrine of the Maturity of Chances. This fallacy says that
something is more likely to happen in the future if it hasn't happened
in a while. Conversely, something is less likely to happen in the
future if it has been happening frequently.
In other words, the odds have to even out. If the
7 has been rolling a lot, it will go into hiding for a while so
it makes the right probability. If you don't see too many red numbers,
or even number, or low numbers on the roulette tracker board, the
ball is going to land on more black, odd, or high numbers in the
future so that red and black, odd and even, and high and low each
occur about 50% of the time. If your slot or video poker machine
hasn't hit in a while, it's bound to hit soon. As I always say,
even the tightest machine in the world pays its jackpot once in
a while.
Let's assume that this fallacy is true. You're playing
your favorite machine and it's been so cheap that you're considering
moving it from your favorites list to a different list. But you
decide to stick with it because a cold streak has to be followed
by a hot streak in order for things to even out.
Let me ask you this question: How do you know that
the cold streak you're in now isn't balancing out a prior hot streak?
The fallacy assumes that whatever is happening now is taking things
out of balance and it will be followed by an opposite action to
restore the equilibrium. What happens if what is happening is actually
restoring the balance? What is supposed to happen in the future
then?
Fortunately, we don't have to deal with these puzzles
because the fallacy is a fallacy. It's not true. But I can see where
people have problems believing that it isn't true because, from
a certain perspective, it is true.
The odds will even out in the long run, but only
in the long run. In the short run, anything can and does happen.
But these short-run anomalies, hot or cold, are just drops in the
bucket when you look at the millions of spins or hands played on
a typical slot machine and the millions of throws of the dice at
a craps table.
Notice that I didn't mention blackjack at all. Slots,
video poker, and craps are all independent trial games. What has
happened in the past does not affect what will happen in the future.
Blackjack is not an independent trial game. The odds constantly
shift as cards are dealt out of the shoe. In blackjack, what has
happened in the past does affect what will happen in the future.
That's the basis for card counting.
I can see another reason why some people have difficulty
believing this fallacy is false. It sometimes seems like I'm contradicting
myself. On the one hand, I'm saying that I can't tell you what's
going to happen on a slot or video poker machine. Then, on the other
hand, I'm saying that certain video poker and slot machines pay
back more than others in the long run. If everything is random,
how can I say with near certainty what a machine will pay back?
Let's say I have a paper bag and I put three ping
pong balls--one white, one red, and one blue--in it. I close my
eyes, reach into the bag, pull out a ball at random, note its color,
replace it in the bag, and repeat the process. I keep track of the
total number of times I've picked each color of ping pong ball.
Some weird things may happen in the short run, especially
if I hit a long streak or dearth of one ball early on, but as I
do this more and more, the number of times I've seen a color divided
by the total number of draws will get closer and closer to 1/3.
If you put in many more colored ping pong balls--each
color corresponding to a particular outcome--and you put in the
appropriate number of each color to represent the likelihood of
getting that outcome, you have a low-tech representation of a slot
or video poker machine. If you perform our experiment with that
paper bag, your results will keep getting closer and closer to the
population in the bag.
I can't tell
you what's going to happen next, but I can tell you what the overall
results will look like.
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