The origins of Bayes’ insights: a puzzle

In English speaking countries the Rev. Thomas Bayes is credited with the notion that all kinds of uncertainty can be represented by numbers, such as P(X) and P(X|Y), that can be combined just as one can combine probabilities for gambling (e.g. Bayes’ rule).

You are told that one of these is true:

  1. Bayes was in the  habit of attending the local Magistrates Court and making an assessment of the defendant’s guilt based on his appearance, and then comparing it with the verdict.
  2. Bayes performed an experiment in which he blindly tossed balls on to a table while an assistant told him whether the ball was to the right or left of the original.

Assign probabilities to these statements. (As usual, I’d be interested in your assumptions, theories etc. If you don’t have any, try here.) 

More similar puzzles here.

Dave Marsay

The Sultan’s daughters

The IMA website has the following puzzle:

A sultan has 100 daughters. A commoner may be given a chance to marry one of the daughters, but he must first pass a test. He will be presented with the daughters one at a time. As each one comes before him she will tell him the size of her dowry, and he must then decide whether to accept or reject her (he is not allowed to return to a previously rejected daughter). However, the sultan will only allow the marriage to take place if the commoner chooses the daughter with the highest dowry. If he gets it wrong he will be executed! The commoner knows nothing about the distribution of dowries. What strategy should he adopt?

You might want to think about it first. The ‘official’ answer is …

 

 

 

 

 

One strategy the commoner could adopt is simply to pick a daughter at random. This would give him a 1/100 chance of getting the correct daughter. [But] the probability of the commoner accepting the daughter with the highest dowry is about 37% if he rejects the first 37 daughters and then chooses the next one whose dowry is greater than any he’s seen so far. This is a fraction 1/e of the total number of daughters (rounded to the nearest integer) and is significantly better than just choosing at random!

My question:

Given that the sultan knows what dowry each daughter has, in which order should he present the daughters to minimise the chance of one of them having to marry the commoner? With this in mind, what is the commoner’s best strategy? (And what has this to do with the financial crisis?)

See also

More puzzles.

Dave Marsay