Probability and rationality — trickier than you might think

1 Jul, 2018 at 23:19 | Posted in Statistics & Econometrics | 3 Comments

The Coin-tossing Problem

My friend Bengt says that on the first day he got the following sequence of Heads and Tails when tossing a coin:
H H H H H H H H H H

And on the second day he says that he got the following sequence:
H T T H H T T H T H

184bic9u2w483jpgWhich day-report makes you suspicious?

Most people I ask this question says the first day-report looks suspicious.

But actually,​ both days are equally probable! Every time you toss a (fair) coin there is the same probability (50 %) of getting H or T. Both days Ben makes equally many tosses and every sequence is equally probable!

The Linda Problem

Linda is 40 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.

Which of the following two alternatives is more probable?

A. Linda is a bank teller.
B. Linda is a bank teller and active in the feminist movement.

‘Rationally,’ alternative B cannot be more likely than alternative A. Nonetheless Amos Tversky and Daniel Kahneman reported — ‘Judgments of and by representativeness.’ In D. Kahneman, P. Slovic & A. Tversky (Eds.), Judgment under uncertainty: Heuristics and biases. Cambridge, UK: Cambridge University Press 1982 — that more than 80 per cent​ of respondents said that it was.

Why do we make such ‘irrational’ judgments in both these cases? Tversky and Kahneman argued that in making this kind of judgment we seek the closest resemblance between causes and effects (in The Linda Problem, between Linda’s personality and her behaviour), rather than calculating a ​probability, and that this makes alternative B seem preferable. By using a heuristic called representativeness, statement B in The Linda Problem seems more ‘representative’ of Linda based on the description of her, although from a probabilistic point of view it is clearly less likely.

3 Comments

  1. If our competing hypotheses are that the coin if fair or not, then the former but not latter leads to a strong updating in favor of unfairness.

    Comment by Kieran Latty— 10 Sep, 2018 #

  2. The coin toss suspicion is not irrational!

    Yes it’s quite possible that 10 heads is purely chance, but it’s still a roughly 1:1000 chance, so one would be justified in thinking there is something dubious going on. The other outcome is just the same … but only if you specify that particular run before you start.

    But 10 tosses is a fairly short run. If you had posited a run of, say, 20 heads, then the argument breaks down completely. Probability theory would suggest that the 21st toss would be a 50:50 bet, but no gambler would believe you because the run of 20H is a~ million to one sequence.

    It’s therefore a close certainty that the tosses are not fair. Read The Black Swan where Taleb discusses this.

    Comment by Peter Baker— 3 Jul, 2018 #

  3. Lars, your first all depends on what you mean by ‘suspicious’. I defend ‘most people’ at https://djmarsay.wordpress.com/notes/puzzles/suspicious-coins/ . I would welcome any enlightenment on what you think suspicion is, or on my suggestion. Ta.

    The Linda example is more complicated, but here too I have more sympathy with the common people than with the academics. Doesn’t the answer depend on whether you accept (or even know about) the subtleties of the somewhat artificial academic conventions?

    Comment by Dave Marsay— 2 Jul, 2018 #


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