# The Gambler's Fallacy, the Fallacy of the Maturity of Chances

Topics: Gambler's fallacy, Polycephaly Pages: 9 (3171 words) Published: October 25, 2011
Gambler's fallacy

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Gambler's fallacy

Gambler's fallacy

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An example: coin-tossing
The gambler's fallacy can be illustrated by considering the repeated toss of a fair coin. With a fair coin, the outcomes in different tosses are statistically independent and the probability of getting heads on a single toss is exactly 1⁄2 (one in two). It follows that the probability of getting two heads in two tosses is 1⁄4 (one in four) and the probability of getting three heads in three tosses is 1⁄8 (one in eight). In general, if we let Ai be the event that toss i of a fair coin comes up heads, then we have,

Simulation of coin tosses: Each frame, a coin is flipped which is red on one side and blue on the other. The result of each flip is added as a colored dot in the corresponding column. As the pie chart shows, the proportion of red versus blue approaches 50-50 (the Law of Large Numbers). But the difference between red and blue does not systematically decrease to zero.

. Now suppose that we have just tossed four heads in a row, so that if the next coin toss were also to come up heads, it would complete...

References: [1] Lehrer, Jonah (2009). How We Decide. New York: Houghton Mifflin Harcourt. p. 66. ISBN 978-0-618-62011-1. [2] Colman, Andrew (2001). "Gambler 's Fallacy - Encyclopedia.com" (http:/ / www. encyclopedia. com/ doc/ 1O87-gamblersfallacy. html). A Dictionary of Psychology. Oxford University Press. . Retrieved 2007-11-26. [3] O 'Neill, B. and Puza, B.D. (2004) Dice have no memories but I do: A defence of the reverse gambler 's belief. (http:/ / cbe. anu. edu. au/ research/ papers/ pdf/ STAT0004WP. pdf). Reprinted in abridged form as O 'Neill, B. and Puza, B.D. (2005) In defence of the reverse gambler 's belief. The Mathematical Scientist 30(1), pp. 13–16. [4] Tversky, Amos; Daniel Kahneman (1974). "Judgment under uncertainty: Heuristics and biases". Science 185 (4157): 1124–1131. doi:10.1126/science.185.4157.1124. PMID 17835457. [5] Tversky, Amos; Daniel Kahneman (1971). "Belief in the law of small numbers". Psychological Bulletin 76 (2): 105–110. doi:10.1037/h0031322. [6] Tversky & Kahneman, 1974. [7] Tune, G.S. (1964). "Response preferences: A review of some relevant literature". Psychological Bulletin 61 (4): 286–302. doi:10.1037/h0048618. PMID 14140335. [8] Tversky & Kahneman, 1971. [9] Gilovich, Thomas (1991). How we know what isn 't so. New York: The Free Press. pp. 16–19. ISBN 0-02-911706-2.
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Article Sources and Contributors
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