This is a problem that I thought I 'solved' many years ago.(adsbygoogle = window.adsbygoogle || []).push({});

In actual fact there are many things about it that are not clear to me, and I would like to hear your opinion, please.

Very briefly, there was this TV programme where a (supposedly psychic) guy had to match 5 (husband-wife) couples, obviously without knowing anything about them.

My interpretation was: if we haveNdistinct objects (say, the letters A, B, C, D, E) that must be placed in one specific order, how likely are we to placek= 0, 1, 2 ...Nof them correctly by choosing the order randomly?

I was pretty sure the problem had long been studied, but I wanted to have some fun figuring it out for myself.

I found quite easily the probability to guessat leastspecific couples correctly, regardless of what happens to the other couples (that'sk(N-k)!/N!,I believe, in this case withN=5).

However, the original problem was to find the probabilityP(k,N)to guessk(no matter which) couples and not the remaining. Things got a bit tougher then, at least for my limited maths skills, and sums with factorials and alternating signs started appearing, but in the end I seemed to find a formula that accounted for the explicitly enumerated cases:N-k

[itex]P(k,N) = \frac{1}{k!} \sum_{i=0}^{N-k}\frac{(-1)^{i}}{i!}[/itex]

One nice feature of this result is that it correctly tells you that P(N-1,N)=0 (it's impossible to guess 4 couples right and not the 5th one). And it also tells you that for odd values ofN, it is more likely to guess 1 couple right than 0 right, meaning perhaps you're more of a 'psychic' if you get them all wrong than if you guess 1 right(!).

Another nice thing about it (which I can't prove, but so far has worked numerically) is that it sums up to 1:

[itex]\sum_{k=0}^{N}P(k,N) = 1 [/itex]

As required, I believe, because the events fork= 0, 1, 2...Nare mutually exclusive and, taken together, they constitute the whole set if possible outcomes of this experiment, so the probability that at least one of them happens must be 1.

What I wanted to know next (and I did this only a few days ago, years after finding the above formula) was the expected value, i.e. if we did a large number of trials, what would be the average number of correctly guessed couples?

Somewhat to my surprise, I found that, at least for the values ofNI tried (2, 3, 4, 5, 10, 20}, it's always 1:

[itex]\sum_{k=0}^{N}k \cdot P(k,N) = 1 [/itex]

as in: regardless of how many couples there are, on average you're likely to guess only 1 right if you choose at random.

I plotted the probabilities for the above numerical cases, and it seems that, after some oscillations forN=2, 3 and 4, already fromN=5 the curve gets very close to a limiting case where the probabilities ofk= 0 or 1 are always the highest and both very close to 0.36788 (if anything is special about this number, I don't know), and the rest of the curve .

Now, I'm no mathematician - it may be that all of the above is particularly obvious to an expert. But it's not to me.

So here are my questions/doubts:

1. when I found the formula I wanted to check if it was correct, but I could find no website describing it - does anybody know if this kind of problem has a particular name?

2. can the formula be reduced to a closed form?

3. is there a way to prove that the probabilities fork= 0 ...Nsum up to 1, and that the expected value is independent ofNand is also 1?

4. if we wanted to test whether a guy is a psychic (OK, nobody is, but bear with me), what statistical method would we use? Chi squared, based on the expected and observed number of successes in N_{T}trials?

Thank you

L

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# I The statistics of 'psychic challenge'

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