Law of Large Numbers and the Gambler's Fallacy

Date: 05/07/2001 at 00:58:11
From: Elvino Basilio
Subject: Dice probabilities

Dr. Math,

To throw three dice three times together, is the same than to throw 
nine dice one time?

We have the same average for to get one same number in the two cases? 
Where and how different is it?

[Editor's interpretation:
  Should you get the same total, on average, when you make three 
  throws of three dice each as when you throw nine dice at once?]

Thank you,
Basilio

Date: 05/08/2001 at 14:06:52
From: Doctor TWE
Subject: Re: Dice probabilities

Hi Elvino - thanks for writing to Dr. Math.

The sum, the average, and the probabilities of getting a particular 
value on one die or more aren't affected by whether the dice are 
rolled one at a time, in groups of 3, or all 9 at once. These are 
called "independent events," and the order in which they happen 
doesn't affect the outcome.

I hope this helps. If you have any more questions, write back.

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

Date: 05/09/2001 at 18:01:50
From: Elvino Basilio
Subject: Dice probabilities

Ola!

So, for to get one six (or other number), when I throw three dice all 
together I have 42,12% possibilities. If I throw the same dice, in the 
same way, again and again I will start, always, with the same 
percentage!

[Editor's interpretation:
  So when I throw three dice, I have a 42.12% chance of rolling a 
  total of 6. If I throw the dice over and over, I should always 
  have the same 42.12% chance of rolling a total of 6. Is this 
  right?]

But, there are one law that said us more I throw more possibilities I 
have for to obtain a certain number. How works these law of 
probabilities? If works...

[Editor's interpretation:
  But there is a law in probability that states that the more often 
  I throw the dice, the closer the experimental probability should 
  come to the theoretical probability. How does that work, if it's 
  true?]

Thank you,
Basilio

Date: 05/10/2001 at 13:44:43
From: Doctor TWE
Subject: Re: dice probabilities

Hi Basilio - thanks for writing back.

>So, for to get one six (or other number), when I throw three dice all 
>together I have 42,12% possibilities. If I throw the same dice, in 
>the same way, again and again I will start, always, with the same 
>percentage!

Yes, that's exactly right.

>But, there are one law that said us more I throw more possibilities I 
>have for to obtain a certain number. How works these law of 
>probabilities? If works...

There is something called the Law of Large Numbers (or the Law of 
Averages) which states that if you repeat a random experiment, such as 
tossing a coin or rolling a die, a very large number of times, your 
outcomes should on average be equal to (or very close to) the 
theoretical average.

Suppose we roll three dice and get no 6's, then roll them again and 
still get no 6's, then roll them a third time and STILL get no 6's. 
(This is the equivalent of rolling nine dice at once and getting no 
6's, as we discussed in the last e-mail; there's only a 19.38% chance 
of this happening.) The Law of Large Numbers says that if we roll them 
500 more times, we should get at least one 6 (in the 3 dice) about 212 
times out of the 503 rolls (.4213 * 503 = 211.9).

This is *not* because the probability increases in later rolls, but 
rather, over the next 500 rolls, there's a chance that we'll get a 
"hot streak," where we might roll at least one 6 on three or more 
consecutive rolls. In the long run (and that's the key - we're talking 
about a VERY long run), it will average out.

There is also something called the Gambler's Fallacy, which is the 
mistaken belief that the probability on the next roll changes because 
a particular outcome is "due." In the example above, the probability 
of rolling at least one 6 in the next roll of the three dice (after 
three rolls with no 6's) is still 42.13%. A (non-mathematician) 
gambler might think that the dice are "due," that in order to get the 
long-term average back up to 42%, the probability of the next roll 
getting at least one 6 must be higher than 42%. This is wrong, and 
hence it's called "the Gambler's Fallacy."

I hope this helps. If you have any more questions or comments, write 
back again.

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