Difficulties in Thinking about Random Processes

• Last time we talked about problems in manipulating probabilities. Today we turn to a related issue - the difficulty of developing good intuitions about stochastic (random) processes.

• We tend to secretly smuggle in notions of pattern, or order, or causality that don’t belong.

Expecting Too Much Regularity in Strings of Coin Tosses

• If there is any random process that should be simple to understand and familiar to us it is surely the behavior of fair coins. Even little children learn that coin tossing is a fair way to resolve disputes about which team bats first.

• Yet, even here, we goof up. When asked to write down some random sequences of coin tosses (without actually tossing a coin) we tend to make the ratio of Heads/Tails come out exactly equal to one and we also include way too many HTHT alternating mini-strings in our sequences.

Gambling Fallacies

• Sometimes after a striking series of Reds at the roulette wheel, people have a strong preference for betting on Black because “things have got to average out in the long run”.

• Here they are confused about the Law of Large Numbers: If a random phenomenon is repeated many times independently, the mean value of the actually observed outcomes does approach the expected value, but that does not mean that there is an underlying mechanism to even things out.

• Other times they strongly opt for Red because “Red is hot tonight on this wheel”. Here again this makes no sense if the spins of the wheel are independent and the wheel is fair.

The Fallacy of Believing in the “Law of Small Numbers”

• “The mean IQ of the population of eighth graders in a city is known to be exactly 100. You have selected a random sample of 50 children for a study. The first child tested has an IQ of 150. What do you expect the mean IQ to be for the whole sample?” (Plous, p. 113)

• People tend to overestimate the degree to which small samples resemble the population as a whole. (Hence the joke about the “law of small numbers”.

• A small sample can easily be atypical of the population as a whole - which is why statisticians like large samples!

The Hospital Example

• “A certain town is served by two hospitals. In the larger hospital about 48 babies are born each day; in the smaller about 16. As you know, about 50% of all babies are boys. The exact percentage, however, varies from day to day - sometimes higher, sometimes lower.

• For a period of 1 year, each hospital recorded the days on which more than 60% of the babies born were boys. Which hospital do you think recorded more such days?”

• Which hospital do you think is more likely to have at least 1 day in which 80% of the babies were boys?

Galton(1889) on the Relationship between the Characteristics of Parents and Children

• Sir Francis Galton, a cousin of Charles Darwin, wrote a book called Natural Inheritance in which he collects scads of data which he thought showed that height, eye color, and artistic temperament (!) are inherited.

• But although he found strong correlations between the attributes of parents and children, he also uncovered a surprising feature in his data:

Quote from Galton

“However paradoxical it may appear at first sight, it is theoretically a necessary fact, and one that is clearly confirmed by observation, that the Stature of the adult offspring must on the whole, be more mediocre than the stature of their parents; that is to say, more near to the M [median or mid-stature] of the general Population.”