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Fair Chance: Three Things, One Coin to Flip

```Date: 03/12/2003 at 09:51:47
From: Marco
Subject: Coin flipping

How can I choose with equity (p = 1/3) among 3 things (door, person,
etc.) having only a (perfect) coin and not infinite time?

Or: How can I generate an event with probability p = 1/3 with one
perfect coin in a finite number of tosses?
```

```
Date: 03/13/2003 at 12:03:14
From: Doctor Achilles
Subject: Re: Coin flipping

Hi Marco,

Thanks for writing to Dr. Math.

Here's my solution (I am pretty sure this is the only way to actually
do it, but there may be a more elegant solution possible):

You need to flip your coin twice.  The possible outcomes are:

HH
HT
TH
TT

If you get HH, take the first option. If you get HT, take the second,
if you get TH, take the third. If you get TT, repeat the process from
the start.

In this strategy, the probabilities for each of the outcomes are
equal. You have a 1/4 chance of having to repeat once, a 1/16 chance
of having to repeat twice, a 1/64 chance of having to repeat 3 times,
etc.

I cannot guarantee that this strategy will terminate in a finite
amount of time, but in reality it should terminate fairly quickly.
Most importantly, you do have exactly a 1/3 chance of each outcome
when you do reach a termination point.

Hope this helps. If you have other questions or you'd like to talk

- Doctor Achilles, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Permutations and Combinations
High School Probability

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