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:
- Toss the coin twice.
- If the results match, start over, forgetting both results.
- 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.
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Famous quotes containing the words fair, results, biased and/or coin:
“Monday’s child is fair in face,
Tuesday’s child is full of grace,
Wednesday’s child is full of woe,
Thursday’s child has far to go,
Friday’s child is loving and giving,
Saturday’s child works hard for its living;
And a child that is born on a Christmas day,
Is fair and wise, good and gay.”
—Anonymous. Quoted in Traditions, Legends, Superstitions, and Sketches of Devonshire, vol. 2, ed. Anna E.K.S. Bray (1838)
“Life and language are alike sacred. Homicide and verbicide—that is, violent treatment of a word with fatal results to its legitimate meaning, which is its life—are alike forbidden.”
—Oliver Wendell Holmes, Sr. (1809–1894)
“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)