Chance

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Union Law - Probability that at least one of two events will occur is sum of probability of each minus the probability of both. E.g. chance of drawing either a spade or a high card(face card or ace) or both = 1/4 + 16/52 - 4/52 = 0.48

Probability P of A or B or C ... = 1 - (P not A) * (P not B) * (P not C) ...
E.g. if Yankees favored 3/5 and 4/7, odds of winning one of these games is (3/5+4/5)-(3/5*4/5)=7/5-12/25=23/25

Paradox of the Chevalier de Mere: Believed odds of getting at least one 1 in four rolls of a die were 4 x 1/6 = 4/6, and odds of getting two 1s in 24 (6x4) rolls were the same. Actually they are 1 - (5/6)^4 = 0.5177, and 1 - (35/36)^24 = 0.4914. This was determined by his friend Blaise Pascal with Pierre de Fermat, who started modern probability theory. Error obvious when you consider chances of getting at least one head in two coin tosses (i.e. obvious not 2 * 1/2).

Monkeys Typing Hamlet - Assuming 30 character keyboard, ignoring spaces and caps, odds = 1/30^142,943 = 0.0<200,000 zeros>1

Benford's Law - Also called the first-digit law, states that in lists of numbers from many real-life sources of data, the leading digit is distributed in a specific, non-uniform way. According to Benford's law, the first digit is 1 almost one third of the time, and larger numbers occur as the leading digit with less and less frequency as they grow in magnitude, to the point that 9 is the first digit less than one time in twenty. This is based on the observation that real-world measurements are generally distributed logarithmically, thus the logarithm of a set of real-world measurements is generally distributed uniformly. See Rex Swain's page

Coin flip trick based on Benford's Law: Tell students to either flip a coin 200 times or just fake the results; any series that doesn't have a run of 6 is declared a fake.

Which is greater: The number of six-letter words with n as the fifth letter or the number of six-letter words ending in -ing. (Most people say -ing) ( DW p28)
Kahneman and Tversky - Experiment ranking probability rank ( DW p22-23)

Trial experiment: Same evidence, but vivid and pallid versions (e.g. garbage truck owner "What, you think we should paint them pink?" Majority side with vivid evidence. ( DW p29)

The Birthday Problem: With 23 random people, there is a 50% percent chance that two of them share the same birthday. With 56 people that number jumps to a 95% chance.

Rats outperform humans in guessing whether a red light or green light will flash in random ordering of 75% red and 25% green. Rats will simply go with more common color every time whereas humans will try to detect pattern. In split brain patients, the isolated right brain will emulate rats. ( DW p6)

"Probability and chance and the interest human beings have had in them predate historical times; Dice made of animal bones have been found and dated to the Neolithic period, more than six thousand years ago. They look remarkably similar to modern dice. In other words, at about the same time the earliest farming societies were being formed, man started playing ur-craps." (CGG pIX)

"Surprisingly, the ancient Greek mathematicians, Pythagoras, Euclid and others, did not spend any time thinking about probability theory." (CGG pXI)

To have a 75% chance of the better team winning when one team is ten percent better than the other (i.e. the probability is 55/45 for better/worse team to win), they would need to play a 45-game series. (Leonard Mlodinow, Town Hall lecture 5/14/2008)

Bill Miller - Fund manager beat the S&P 15 years in a row (by chance, likelihood about 75% given 6,000 fund managers over 40 years) ( DW p186,188)

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