Do Prior Outcomes Affect Probabilities of Future Ones?

Date: 05/26/2008 at 04:31:06
From: Konstantinos
Subject: Are possible outcomes relative with previous results

What I want to know is if in matters of luck, such as games of dice, 
or lotteries, or flipping coins, the future outcomes have any 
relativity with past results.  What I mean in each case is:  If I 
flip a coin and get tails, don't I have bigger or smaller 
possibilities to get heads on the next roll?  I mean I know that the 
possibility is always 1/2 but since I have already thrown the coin 5 
times and rolled 5 tails there aren't possibilities that in the next 
throws there will be heads?  The same question goes for dice rolls, 
and for lotto numbers.  If a number has come more times than others, 
isn't it possible that for a limited amount of coming times, numbers 
that haven't come yet, will start showing more?

What I find most confusing is that the relation between probabilities 
and past possibilities, and the outcome in real life.  How can a coin 
come tails 5 or six times in a row when the possibility is always 
1/2?  Are probabilities only theoretical?

I tried noting down results of different trials of luck (dice, past 
lotteries, coins) but in the end they don't seem to make true to any 
theory I have heard.  I would like to see the magic of numbers in 
real life and on this subject and how it works as to prove a theory.  
Do I have to toss the coin 10,000 times?  And if I do will I see 
heads coming in a row after 10 subsequent tails?

Date: 05/26/2008 at 22:48:43
From: Doctor Peterson
Subject: Re: Are possible outcomes relative with previous results

Hi, Konstantinos.

What you are suggesting is called the Gambler's Fallacy--the WRONG 
idea that future results of a random process are affected by past
results, as if probability deliberately made things balance out.  The
law of large numbers says that it WILL balance out eventually; but
that does not happen by changing the probabilities in the short term.
The long-term balance just swamps the short-term imbalance.

If you think about it, what could cause a coin to start landing heads
up more often after a string of tails?  There is no possible physical
cause for this; the coin has no memory of what it, or any other coin,
previously did.  And probability theory does not make things happen
without a physical cause; it just describes the results.

If I tossed a coin and it landed tails 5 times, I would just recognize
that that is a perfectly possible (and not TOO unlikely) thing to
happen.  If I got tails 100 times, I would NOT expect the next to be
heads; I would inspect the coin to make sure it actually has a heads
side!  An unlikely string of outcomes not only does not mean that the
opposite outcome is more likely now; it makes it LESS likely, because
it suggests statistically that the basic probability may not be what I
was originally assuming.

You wrote:

>What i find most confusing is that the relation between 
>probabilities and past possibilities, and the outcome in real life.  
>How can a coin come tails 5 or six times in a row when the 
>possibility is always 1/2?  Are probabilities only theoretical?

Probabilities are theoretical, but have experimental results, IN THE
LONG RUN.  The law of large numbers says that, if you repeat an
experiment ENOUGH TIMES, the proportion of times an event occurs will
PROBABLY be CLOSE to the predicted theoretical probability. 
Probability can't tell you what will happen the next time, but it does
predict what is likely to happen on the average over the next, say,
million times.  If you started out with ten tails in a row, you will
not necessarily get ten heads in a row at any point, or even more
heads than tails in the next few tosses; you will just get enough
heads in the next million to keep it balanced.

In particular, if you were to throw five coins at a time (to make it
easier than throwing the same coin five times in a row), and do that
1000 times, you would expect that you would get 5 tails about 1/32 of
those times (since the probability of all five being tails is 1/2^5).
That's about 31 times!  So it's not at all unreasonable to expect that
it will occur once in a while.  

On the other hand, the probability of getting 100 tails in a row is
1/2^100, or 1/1,267,650,600,228,229,401,496,703,205,376, which makes
it very unlikely that it has ever happened in the history of the
world, though it could!

>I would like to see the magic of numbers in real life and on this
>subject and how it works as to prove a theory.  Do I have to toss the
>coin 10,000 times?  And if I do will I see heads coming in a row
>after 10 subsequent tails?  

Again, probability can't tell you what WILL happen, specifically; it
is all about unpredictable events.  But if you tossed a coin 1000 sets
of 10 times, on the average one of those is likely to yield 10 tails.
(The probability is 1/2^10 = 1/1024.)  The probability of some ten in
a row out of 10,000 tosses is a little bigger, but that's harder to
calculate.

If you have any further questions, feel free to write back.


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

Date: 05/27/2008 at 02:57:11
From: Konstantinos
Subject: Thank you (Are possible outcomes relative with previous results)

Thanks a lot for the immediate reply, you helped me a lot to
understand a few things I kept questioning myself over and over again.
I also appreciate the fact that you answered each part of the question
separately. I will definitely keep your site in mind in case I or
someone I know needs any help in the future. Thanks again!