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 Features
Statistically Speaking, It's Probably a Good Game, Part 1: Probability for Game Designers
 Superstition and the Fallacy of Equipartition — aka “The Gambler’s Fallacy”
One of the most common and widespread thought errors that people make concerning probability is blurring the line between independent and related events. This typically takes one of the following forms:
 BAD THINKING AHEAD!
Mistake 1: Believing that a “5” is less likely than normal to appear again because the last dice roll was a “5”.
Mistake 2: Believing that a “6” has a very high chance of being thrown because 10 rolls have gone by without a “6” being thrown. Dressed in another outfit, this is believing that “red” is due on a roulette wheel because it has been several spins since the last “red” hit.
Mistake 3: After flipping a coin 10 times and getting 8 heads and 2 tails, believing that the next 10 flips will have more tails then heads in order to “even out.”
These all loosely fall under the appropriately well-named “Gambler’s Fallacy.” Basically, this is just the name for confusing independent and related events. Another name for this fallacy is “I just lost all my money at roulette because the Laws of Probability defied me Fallacy.” It is closely related to the lesser-known “Why do casinos allow me to keep a written log of the recent roulette spins — surely they know that I’ll be able to figure out the pattern and beat the wheel Fallacy?” (Note: that last fallacy typically is followed quickly by the previous one.)
Don’t fall into these traps! Rolling a die multiple times or spinning a roulette wheel are independent events, pure and simple. Let’s examine each of the above mistakes more closely:
Mistake 1: The chance of rolling a “5” on a d6 is 1/6 = 16.7% This never changes. It doesn’t matter if you’ve thrown eight “5”s in a row or haven’t seen a “5” since Gilligan’s Island premiered. 16.7% is still the magic number. “Dice don’t have memory” is a common phrase overheard...and it’s correct!
Mistake 2: Same as above. The chance of rolling a “6” or hitting “red” has absolutely nothing to do with the rolls or spins that came before. Roulette wheels don’t have memories either (unless they are actually magnetized and “Vinnie the Spinnie” is making sure that your number never comes up).
The most common argument that people make for mistakes 1 and 2 goes something along these lines:
GAMBLER’S FALLACY, scene 24b:
INT. HALLWAY — NIGHT
The moon shines in, bathing the office in silvery luminescence.
Game developers roam the halls like zombies, crunching for a
milestone. Or crunching human bones...
Mr. Faulty bumps into Probability Stuffshirt; both jolt out of
their zombie-like zombified zombie trances.
MR. FAULTY
(wryly)
Surely you agree that rolling
three “5”s in a row is unlikely.
PROBABILITY STUFFSHIRT
Yes. The chances are 1/6 x 1/6 x
1/6 = 1/216 = less than 0.5%, to
be exact.
MR. FAULTY
Well I’ve just rolled two “5”s in
a row, so my chance of rolling
another “5” right now is surely
very, very small! Less than
normal!
PROBABILITY STUFFSHIRT
Actually, your chances of rolling
another “5” right now are 1 in 6 =
16.7%.
MR. FAULTY
How can I have a 16.7% chance if
there is only a 0.5% chance of
rolling three 5’s in a row? I’ve
got you!
A well-dressed, distinguished looking NARRATOR emerges from a
small space behind the water cooler. He addresses the camera
directly.
NARRATOR
Mr. Faulty is making a patently
bad but sadly common argument.
Let’s hear why...
PROBABILITY STUFFSHIRT
Not so fast. You have already
rolled two “5”s in a row. The
chances of that were a mere 1/6 x
1/6 = 1/36 = less than 3%. You’ve
already done the hard part. Now
all you have to do is roll another
“5”, and you have a 1/6 chance to
do it. If you succeed, then that
completes the 1/6 x 1/6 x 1/6 =
1/216 = 0.5% Take that!
MR. FAULTY
(pointing wryly)
What could that be over there!
(escapes to cubicle)
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