Gambler's Fallacy

Date: 03/14/2003 at 14:13:18
From: Kevin
Subject: Lottery: Betting same number vs randomly selecting a number.

A current co-worker and I are in a friendly disagreement about the 
probability of selecting the winning number in any lottery, say pick 
5. He states that he would rather bet the same set of five numbers 
every time for x period of time, but I insist that the probability is 
the same if you randomly select any set five numbers for the same 
period of time. The only assumption we make here is betting one set of 
numbers on any given day. Who is correct?  

I tried explaining to him that the probability of betting on day one 
is the same for both of us. On day two it is the same. On day three it 
is the same, etc. Therefore the sum of the cumulative probabilities 
will be the same for both of us.

Thank you for your anticipated response.

Date: 03/15/2003 at 03:29:05
From: Doctor Wallace
Subject: Re: Lottery: Betting same number vs randomly selecting a 
number.

Hello Kevin,

You are correct. If you have the computer randomly select a different 
set of 5 numbers to bet on every day, and your friend selects the same 
set of numbers to bet on every day, then you both have exactly the 
same probability of winning.

Tell your friend to think of the lottery as drawing with tickets 
instead of balls. If the lottery had a choice of, say, 49 numbers, 
then imagine a very large hat containing 1 ticket for every possible 
combination of 5 numbers.  1, 2, 3, 4, 5;  1, 2, 3, 4, 6;  etc.

On the drawing day, ONE ticket is pulled from the hat. It is equally 
likely to be any of the C(49,5) tickets in the hat. (There would be 
1,906,884 tickets in the hat in this case.)

Since both you and your friend have only ONE ticket in the hat, you 
both have the same chance of winning.

On the next drawing day for the lottery, ALL the tickets are replaced.  
Each lottery draw is an event independent of the others. That is to 
say, the probability of any combination winning today has absolutely 
NO effect on the probability of that or any other combination winning 
tomorrow. Each and every draw is totally independent of the others.

The reason your friend believes that he has a better chance of winning 
with the same set of numbers is probably due to something called the 
"gambler's fallacy." This idea is that the longer the lottery goes 
without your friend's "special" set of numbers coming up, the more 
likely it is to come up in the future. The same fallacy is believed by 
a lot of people about slot machines in gambling casinos. They hunt for 
which slot hasn't paid in a while, thinking that that slot is more 
likely to pay out. But, as the name says, this is a fallacy; pure 
nonsense. A pull of the slot machine's handle, like the lottery draw, 
is completely independent of previous pulls. The slot machine has no 
memory of what has come before, and neither has the lottery. You might 
play a slot machine for 2 weeks without hitting the big jackpot, and 
someone else can walk in and hit it in the first 5 minutes of play.  
People wrongly attribute that to "it was ready to pay out." In 
reality, it's just luck.  That's why they call it gambling.  :)

This used to be a "trick" question on old math tests:

"You flip a fair coin 20 times in a row and it comes up heads every 
single time. You flip the coin one more time. What is the probability 
of tails on this last flip?"

Most people will respond that the chance of tails is now very high.  
(Ask your friend and see what he says.)  However, the true answer is 
that the probability is 1/2.  It's 1/2 on EVERY flip, no matter what 
results came before.  Like the slot machine and the lottery, the coin 
has no memory.

Thanks for writing to Dr. Math. Don't hesitate to write again if you 
need further help with this or another question.

- Doctor Wallace, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 03/17/2003 at 09:50:05
From: Kevin
Subject: Thank you (Lottery: Betting same number vs randomly selecting 
a number.)

Thank you for your prompt and thorough response. You have helped us 
rest this case. Keep up the good work.

Kevin