Currently reading: Ten Great Ideas About Chance.

It can be a bit of a frustrating read – a lot of the chapters about gambling and judgement appear to be poorly explained to me – especially if you’ve ever actually placed a bet on anything (I am far from a regular gambler – I’m not sure I’ve placed a bet on anything in the last decade but I still know how it works).

But it also has lots of exceptionally interesting stuff: not least the revelation that naturally occurring number sets tend to begin with 1 with a proportion of 0.3 whilst fraudsters (including election fixers and presumably scientific experiment fixers) tend to use such numbers 0.111… (1/9) of the time.

So, taking my life in my hands – what about results in my PhD – (read thesis here). I am actually calculating this as I go along – so I *really* do hope it doesn’t suggest fraud!

Anyway – first table, which has a calculated SPECmark figure based on other researchers’ work – so not really my findings, though the calculations are mine.

Numbers are: 35, 22.76, 13.41, 7.26, 3.77, 1.94, 1.01, 0.55, 0.31, 0.20 and 0.14

And (phew) that means 4 out of 11 (0.364) begin with 1 and so looks fraud free on this test.

But as I say, those are my calculations, but they are not my findings at root.

So moving on to some real findings.

First, the maximum total length of time it takes for a simulation to complete a set of benchmarks in simulated cycles: 43361267, 1428821, 1400909, 5760495, 11037274, 2072418, 145291356, 5012280.

Here a whole half are numbers that begin with 1. Hmmmm.

The mean time for the same: 42368816, 1416176, 1388897, 5646235, 10964717, 2026200, 143276995, 4733750.

Again, half begin with one.

What about the numbers within the results – in this case the amount of cycles lost due to waiting on other requests?

The numbers are: 35921194, 870627, 844281, 4364623, 1088954, 1446305, 110996151, 3420685

Now the proportion has fallen to 3/8 (0.375) and I can feel a bit more comfortable (not that the other results suggested I was following the fraudsters’ one-ninth pattern in any case.)

Later I produce numbers for an optimised system. How do they perform?

The maxima now become: 2514678, 1357224, 1316858, 3840749, 10929818, 1528350, 102077157, 3202193.

So the proportion beginning with 1 has actually risen to 5/8.

And the means show a similar pattern. Worse news with the blocks though. They now become: 730514, 775433, 735726, 2524815, 806768, 952774, 64982307, 1775537. So I am left with 1/8 (0.125) starting with 1 – something dangerously close to 1/9.

Can I save myself? I hope so … the figures above are for the first iteration of the system, but when we look at subsequent iterations a different pattern (for the blocks) emerges: 130736, 0, 0, 1612131, 97209, 232131, 64450433, 1117599.

This is now back to 3/8.

Of course, I didn’t cheat and I also suspect (I would, wouldn’t I?) that the block count is a better measure of whether I had or not – because the overall execution time of the benchmarks is, in some sense, a function of how long their execution path is and effectively that is determined – the blocking in the system is an emergent property and if I faked the whole thing I’d have less to go on and be much more likely to make it up.

Well, that’s my story and I’m sticking to it.

I think you are referring to Benford’s Law. Per the Wikipedia page on it (https://en.wikipedia.org/wiki/Benford%27s_law), some distributions are expected to obey it while others are expected not to. Personally, I would be miffed I were using a pseudorandom number generator that kowtowed to Benford.

I imagine you checked this mostly as a lark, but it raises an interesting question. In the context of your results, which would be more incriminating, adhering to Benford’s Law or violating it? 🙂

Yes, it very much looks like it – though the book doesn’t discuss it in those terms.

I started off feeling very pleased that the first set of results worked and then got a bit worried I had opened myself up to all sots of trouble – so was pleased when I found a way to escape from the trap I had set myself.