Fair Coin - Fair Results From A Biased Coin

Fair Results From A Biased Coin

If a cheat has altered a coin to prefer one side over another (a biased coin), the coin can still be used for fair results by changing the game slightly. John von Neumann gave the following procedure:

  1. Toss the coin twice.
  2. If the results match, start over, forgetting both results.
  3. If the results differ, use the first result, forgetting the second.

The reason this process produces a fair result is that the probability of getting heads and then tails must be the same as the probability of getting tails and then heads, as the coin is not changing its bias between flips and the two flips are independent. By excluding the events of two heads and two tails by repeating the procedure, the coin flipper is left with the only two remaining outcomes having equivalent probability. This procedure only works if the tosses are paired properly; if part of a pair is reused in another pair, the fairness may be ruined. Also, the coin must not be so biased that one side has a probability of zero.

Read more about this topic:  Fair Coin

Famous quotes containing the words fair, results, biased and/or coin:

    “While I believe that with a fair election in the South, our electoral vote would reach two hundred, and that we should have a large popular majority, I am yet anxious, as you are, that in the canvassing of results there should be no taint of dishonesty.”
    —Rutherford Birchard Hayes (1822–1893)

    “A distinction of property results from that very protection which a free Government gives to unequal faculties of acquiring it.”
    —James Madison (1751–1836)

    “Scientists are human—they’re as biased as any other group. But they do have one great advantage in that science is a self-correcting process.”
    —Cyril Ponnamperuma (b. 1923)

    “The oft-repeated Roman story is written in still legible characters in every quarter of the Old World, and but today, perchance, a new coin is dug up whose inscription repeats and confirms their fame. Some “Judæa Capta,” with a woman mourning under a palm tree, with silent argument and demonstration confirms the pages of history.”
    —Henry David Thoreau (1817–1862)