What Makes Events Independent?

Date: 06/03/2002 at 07:28:19
From: Milton Chandradas
Subject: Probability of an event

The probability of getting a head, when tossing a coin, is 0.5.

Supposing I have tossed the coin 10 times, and I have always ended up 
with heads.  Common sense kind of tells you that the probability of 
getting a tail on the 11th attempt is greater.  

Why is the probability of getting a heads still 0.5 on my 11th
attempt?

Date: 06/03/2002 at 09:45:04
From: Doctor Ian
Subject: Re: Probability of an event

Hi Milton,

It's still 0.5 because the previous tosses have nothing to do 
with the next toss.  

To see why this is true, try some thought experiments. 

  1. Suppose after 10 tosses, all heads, you switch to a 
     new coin.  Do you still expect the next toss to have
     a greater probability of coming up tails?  If so, 
     then what is the source of this bias?  It can't be 
     the coin, so it must be you.  

  2. Suppose after 10 tosses, all heads, you hand the coin 
     to someone else, so he can toss it.  Do you still expect
     the next toss to have a greater probability of coming 
     up tails?  If so, then what is the source of this bias?
     It can't be you, so it must be the coin. 

  3. Suppose after 10 tosses, all heads, you put the coin in 
     a drawer for 10 years, and then toss it again. Do you 
     still expect the next toss to have a greater probability
     of coming up tails?  

  4. Suppose 10 people toss 10 different coins, and they all
     come up heads.  An 11th person is about to toss an 11th 
     coin.  Do you expect the toss to have a greater probability
     of coming up tails? 

  5. Suppose I hand you a coin and _tell_ you that I've just
     tossed 10 heads in a row with it.  If you toss it, do you
     expect the toss to have a greater probability of coming
     up heads?  Would it make any difference if I lied about
     the previous 10 tosses?  

In the end, each toss is independent because (as far as we know) 
the universe doesn't have anywhere to store the information about 
previous tosses, which means it has no way to influence the next 
toss based on information about the preceding one(s). 

Does this make sense? 

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

Date: 06/03/2002 at 10:36:48
From: Milton Chandradas
Subject: Thank you (Probability of an event)

Thanks.  Your explanation was quite clear and even I could 
understand.  Thanks once again.