Associated Topics || Dr. Math Home || Search Dr. Math

### What Makes Events Independent?

```Date: 06/03/2002 at 07:28:19
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
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
Subject: Thank you (Probability of an event)

Thanks.  Your explanation was quite clear and even I could
understand.  Thanks once again.
```
Associated Topics:
High School Probability
Middle School Probability

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search