Lecture 10/18: Taking Chances
Notes on taking chances
Probability Theory (Chapter Ten)
I. Gambler’s fallacy and the law of large numbers
A. Examples
B. Law of Large numbers: The difference between the observed value of a sample and its true value will diminish as the number of observations in the sample increases.
Applications
O: after a billion flips of a coin, we counted 48% heads, 52% tails.
H1: Fair coin
H2: coin is weighted towards tails
Which hypothesis is predicted by the law of large numbers?
answer: H2, due to the Law of Large Numbers
O: after ten flips of a coin, we counted 6 heads and 4 tails
H1: Fair coin
H2: Coin is weighted towards heads?
Which hypothesis is predicted by the law of large numbers?
answer: predictively equivalent (H1 = H2)
Random sequences:
A. 1, 1, 1, 1, 1, 1, 2, 1, 1, 2
B. 1, 2, 1, 2, 2, 1, 2, 1, 1, 2
C. 1, 2, 2, 2, 2, 1, 2, 1, 1, 2
D. 1, 2, 2, 2, 1, 2, 1, 2, 2, 1
Which one was generated by a “randomizer” ? A
http://www.randomizer.org—form.htm
Question: Suppose you flip a fair coin three heads in a row. What is the
probability that a head will come up a fourth time?
answer: 1/2
C. Misapplication of law of large numbers
Example 1 and 2:
law of large numbers does not support the idea that a gambler will experience runs of good luck after a run of bad luck. For coins and casino machines the probability of any outcome is independent of the number of trails you have experienced. All bets are off if the trials are dependent rather than fair (but then no one would play at such a casino where the outcomes are rigged).
D. Examples outside of gambling
– Hot streaks in basketball: give the ball to the buy who has made a bunch of shots in a row. But, statistically, hitting three or shots in a row is statistically insignificant.
– “market beaters” in fund managing: you swap out of your underperforming funds and into the hot fund. But, given that the market is pretty efficient, past performance is not a good guide to future performance (there will be streaks for any fund over long enough period of time).
II. Common judgements and their fallacious foundations
A. Confirmation bias: you are convinced beforehand that a stock picker or basketball player can “get hot” (due to media attention or your own feelings about the person). So, you ignore the fact that streaks are likely in the short term (given law of large numbers).
B. Over optimism
How often does a college basketball team that is trailing at halftime come back to win?
answer: (less than 20%) (people typically guess 30%60%)
data: 3300 games in NovJan.
Why are we often wrong?
We are optimistic and media gives most attention to comeback victories.
C. Irrationality due to desire to win
Suppose 50% chance of scoring on a twopoint shot. 33% for a threepoint shot. A team is down by two points and it has time for one last shot. What play should the coach call?
answer: if the team makes the twopoint shot, it still has to play overtime, where its chances of winning are 50%. Have to win on two 50% gambles = 25%
overall. So, should go for 3points.
Apply this to stocks: many investors shy away from stocks because of the potential for shortterm sting (like the sting of losing from a 3point shot at the
buzzer). But, in the long run stocks are best investment (over its history).
D. Representative heuristic
question #1 on Tversky teasers.
People tend to say that Hand #2 is much more unlikely than Hand #1. But, each is equally likely in a fair game.
Representative heuristic: Hand #2 is more unimpressive so it is more likely to represent an ordinary hand.
question #2 on “teasers”
89% of students said that it is more likely that Linda was both a bank teller and a feminist than that she was simply a bank teller.
can’t be true: the probability of two things being true can never be higher than the probability that just one of them is true (one is true if both are).

October 22, 2011 at 5:29 pmLecture 10/20: More on Chances « Logic and Reasoning 1200