## A better Bayesian puzzle

Actually, the example of Bayesian reasoning in the post below doesn’t really give the flavor of Tenenbaum’s work. Here’s something a little closer. Abe and Becky each roll a die 12 times. Abe’s rolls come up

1,2,3,4,5,6,1,2,3,4,5,6.

Becky gets

2,3,3,1,6,2,5,2,4,4,1,6.

Both of these results occurs with probability (1/6)^12, or about one in two billion. So why is it that Abe finds his result startling, while Becky doesn’t, when the two outcomes are equally unlikely?

Extra credit: what does this have to do with arguments about intelligent design?

(If you like the extra credit question, you might want to read my colleague Elliot Sober’s papers on the topic, or even buy his book!)

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## 9 thoughts on “A better Bayesian puzzle”

1. Daniel says:

Reminds me of DnD:
Player 1: Woah! I just rolled two 20s in a row! That’s a 1 in 400 chance!
Player 2: Woah! I just rolled a 7, and then a 12! That’s also a 1 in 400 chance…

2. JmSR says:

In Mega Millions, in the last three drawings the MegaBall has been 26. Yet the odds of the same number occuring three times in a row is less than 1:5,000. Yet, everyone needed to know just how unlikely this was… from me… today.

3. […] pointed to maximal entropy methods, still I would like to find the relation… Ah, and see this nice post I found. Published […]

4. Javi says:

What is the probability of this event? I just wrote
an entry in my blog
about Bayesian probability and the temperature of a single configuration in statistical mechanics. And then, following some links from the physicsworld newswire, I reach here… :)

5. Why Abe ought to be startled: Abe’s result is actually far more likely than Becky’s, because it is attributable to cheating.

6. There’s another very reasonable sense in which the first pattern is quite a bit more surprising: the _expected_ number of throws one must wait before the occurrence of Abe’s pattern is 2176828992, while it is “only” 362797056 for Becky’s pattern (i.e.,on average A would appear — consecutively — in an infinite sequence of throws only after waiting about 6 times as long as one needs to wait for B); see

“A Martingale Approach to the Study of Occurrence of Sequence Patterns in Repeated Experiments”, Shuo-Yen Robert Li, The Annals of Probability, Vol. 8, No. 6 (Dec., 1980), pp. 1171-1176

and more specifically Lemma 2.4 which explains how to make this type of computations (for much more general situations).

7. CS says: