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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


  • Law of Unions of Independent Events:
    • 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).


  • Convert Fractional Odds to Implied Probability: Examples:
    • Odds of 4/5 = 5/(4+5) = 5/9 = .5556 = 55.56%
    • Odds of 4/1 = 1/(1+4) = 1/5 = .20 = 20.00%



  • 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.


  • The Availability Bias: Tendency to put too much weight on vivid evidence
    • 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)
      • 2.1 Linda is active in the feminist movement
      • 4.1 Linda is a bank teller and active in the feminist movement
      • 6.2 Linda is a bank teller
      • With only those three choices, 87% said teller/feminist more likely than teller
    • 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.


  • Pattern Seeking
    • 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)


  • If a woman has two children and one is a girl, the chance that the other child is also female has to be 50-50, right? But it's not. Cardano again: The possibilities are girl-girl, girl-boy and boy-girl. So the chance that both children are girls is 33 percent. (Mlodinow)


  • Cardano's Method: "Suppose you want to calculate the likelihood of tossing two coins and coming up with one head. The great 18th-century mathematician Jean Le Rond d’Alembert thought the answer was obvious: there are three possibilities, zero, one or two heads. So the odds for any one of those happening must be one in three. But as Leonard Mlodinow explains in “The Drunkard’s Walk: How Randomness Rules Our Lives,” there are, in fact, four possible outcomes: heads-heads, heads-tails, tails-heads and tails-tails. So there is a 25 percent chance of throwing zero or two heads and a 50 percent chance of throwing just one. In the long run, anyone offering d’Alembert’s odds in a coin-flipping contest would lose his shirt. The key to puzzles like this, Mlodinow writes, is Cardano’s method, named for Gerolamo Cardano, author of the 16th-century “Book on Games of Chance.” To lay the odds for even the simplest-seeming event, one constructs a table, or “sample space,” of all the ways Fortuna’s dice might fall. Trust your instincts instead and you’re bound to go wrong.


  • "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)


  • Sally Clark, convicted in Britain, after two sons died in infancy and bogus statistics presented by expert that chance she was innocent were 1 in 73 million.
  • Which is most likely to happen first in flipping a coin?  HTH or HTT?   Average to HTT is 8, average HTH is 10.   Reason: After you throw an HT, the odds of winning either in the next throw are 50%.  But if you lose playing the HTT game, then you've just thrown an H and you are 1/3 into the pattern again.


  • 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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