Skip to content
All library documents

Generating an Unbiased Outcome from an Unfair Coin

Article Quant Q&A · Author: peakit
Read the original

Summary

The document explains how to obtain an equally likely binary outcome from repeated tosses of a coin with an unknown bias. The method examines tosses in pairs and discards pairs where both results match. For a pair with different results, one order is assigned to one outcome and the reverse order to the other. Because the two orders have equal probability under independent tosses with the same bias, the retained outcomes are balanced.

This procedure is known as the von Neumann corrector. The answers identify the key condition: same-side pairs must be rejected, while opposite-side pairs are used. The method removes bias from the output but does not guarantee an output after a fixed number of tosses; repeated pairs may all match. The explanation assumes the coin's tosses are independent and its probability of heads remains constant. It addresses a probability puzzle rather than a trading strategy, but illustrates a useful sampling technique.

Key ideas

  • Examine coin tosses in pairs and reject pairs that show the same side twice.
  • Map the two possible orders of a mixed pair to opposite outcomes.
  • The mixed orders have equal probability when tosses are independent and the bias is stable.
  • The method can require an unpredictable number of tosses before producing an outcome.

Tags

Statistics
Full text
# Probability - Generating fair outcome using unfair coin


# Probability - Generating fair outcome using unfair coin












I have been thinking a lot about the following puzzle. But, could not arrive at a solution.

Can someone explain me how can you get a fair (equal probability) outcome using only an unfair coin (where unfair means that it will land head with probability p and tails 1-p where p != .5)?

Thanks in advance!

## Answer by G__ (score 7, accepted)

https://quant.stackexchange.com/a/404

As long as you are allowed to discard initial unbalanced results (HH or TT) in favor of balanced results (HT or TH), you can get a fair outcome.

See http://en.wikipedia.org/wiki/Fair_coin for a detailed explanation.

## Answer by vonjd (score 2)

https://quant.stackexchange.com/a/405

You throw the coin twice. When you get heads-tails you count the toss as heads. If the tosses come up tails-heads you count it as tails. You ignore the throws that come up twice the same side. With this method you remove the bias.

This method is also called a von neumann corrector.

## Answer by ObelAnna (score 2)

https://quant.stackexchange.com/a/409

What you are looking for is a Von Neumann Corrector - http://everything2.com/title/von+Neumann+corrector

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.