Another puzzle, courtesy of a mathematics lecture I attended last night. It is  variant of the Birthday ‘Paradox‘. The original ‘paradox’ is that a typical group of people is much more likely to contain two people that share a birthday than most people would think. The variant was where 20 people were asked to pick an integer between 1 and 100 and it was found that two had picked ’42’. The mathematics is the same as for the birthday problem. But is it right?

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Are there any unwarranted assumptions?

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The ‘official’ (Wikipedia) answer to the birthday paradox would be correct if people were randomly selected from a population whose birthdays were uniformly distributed through the year. But are they? This is not a mathematical question, so it cannot be that the answer provided is ‘mathematically’ correct, can it? But one could perhaps say is that the answer provided would be correct if the assumptions were true, and would be approximately correct if they were approximately true – but a sensitivity analysis would be revealing.

The variant brings in greater uncertainties. For example, before the experiment we all guessed the probability. We did much better than the previous audience. Could this be relevant? In any case, why should we expect the guesses to be evenly distributed? Might there not be lucky numbers or other special numbers – such as 42 – that would be chosen?

If numbers were clumped for any reason, the probability of a match significantly increases. I can imagine lots of reasons why numbers should be clumped, but none why they should be anti-clumped, so it seems to me that the ‘official’ probability is actually at the lower end of a range of possible probabilities. Thus if the official probability is 83% I would consider 91% (= (83%+100%)/2) a better guess, and [83%,100%] better still.

The calculations are simpler if we consider the possibility of a match when we toss a coin twice. If P(Heads) = 0.5+e then

P(Match) = (0.5+e)2 + (0.5-e)2 = 0.5 + 2.e2. Thus 0.5 is a lower bound on the probability of a match, provided that coin tosses are independent.

Intelligence-led: Intelligent?

In the UK, after various scandals in the 90s, it seemed that horizon scanning for potential problems, such as the BSE crisis, ought to be more intelligent and even ‘intelligence-led’ or ‘evidence-led’ as against being prejudice or spin-led. Listening to ministerial pronouncements on the horse-meat scandal I wonder if  the current so-called ‘intelligence-led’ approach is actually intelligent.

Suppose that the house next door becomes a refuge for drug-addicts. Which of the following are intelligent? Intelligence-led?

1. Wait until there is a significant increase in crime locally – or until you get burgled – and then up-rate your security.
2. Review security straight away.

In case you hadn’t guessed, this relates to my blog, and the question of what you mean by ‘information’ and ‘evidence’.

Does anyone have a definition of what is meant by ‘intelligence-led’ in this context?

Dave Marsay

P.S. I have some more puzzles on uncertainty.

Coin toss puzzle

This is intended as a counter-example to the view, such as Savage’s, that uncertainty can, in practice, be treated as numeric probability.

You have a coin that you know is fair. A known trickster (me?) shows you what looks like an ordinary coin and offers you a choice of the following bets:

1. You both toss your own coins. You win if they match, otherwise they win.
2. They toss their coin while you call ‘heads’ or ‘tails’.

Do you have any preference between the two bets? Why? And …

In each case, what is the probability that their coin will come up heads?

Dave Marsay

Clarification

In (1) suppose that you can arrange things so that the trickster cannot tell how your coin will land in time to influence their coin, so that the probability of a match is definitely 0.5, with no uncertainty. The situation in (2) can be similar, except that your call replaces the toss of a fair coin.

Other uncertainty puzzles .

Which Car?: a puzzle

Here’s a variation on some of my other uncertainty puzzles:

You are thinking of buying a new car. Your chosen model comes in a choice of red or silver. You are about to buy a red one when you learn that red car drivers have twice the accident rate of those who drive silver ones.

Should you switch, and why?

Dave Marsay

Football – substitution

A spanish banker has made some interesting observations about a football coach’s substitution choice.

The coach can make a last substitution. He can substitute an attacker for a defender or vice-versa. With more attackers the team  more likely to score but also more likely to be scored against. Substituting a defender makes the final score less uncertain. Hence there is some link with Ellsberg’s paradox. What should the coach do? How should he decide?

A classic solution would be to estimate the probability of getting through the round, depending on the choice made. But is this right?

Pause for thought …

As the above banker observes, a ‘dilemma’ arises in something like the 2012’s last round of group C matches where the probabilities depend, reflexively, on the decisions of each other. He gives the details in terms of game theory. But what is the general approach?

The  classic approach is to set up a game between the coaches. One gets a payoff matrix from which the ‘maximin’ strategy can be determined? Is this the best approach?

If you are in doubt, then that is ‘radical uncertainty’. If not, then consider the alternative in the article: perhaps you should have been in doubt. The implications, as described in the article, have a wider importance, and not just for Spanish bankers.

Other Puzzles, and my notes on uncertainty.

Dave Marsay

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?)