Oh no, this weekend they killed Ernie…

Well, actually, not that one. This one…

No, no, no. That one died some time ago. This one…

But don’t worry, here’s Ernie (mark 5)…

Let me explain…

Ernie (Electronic Random Number Indicator Equipment) is the acronym of the random number generator that is used by the government’s National Savings and Investments (NSI) department for selecting Premium Bond winners each month.

Premium bonds are a form of savings certificates. But instead of receiving a fixed or variable interest rate paid at regular intervals, like most savings accounts, premium bonds are a gamble. Each month a number of bonds from all those in circulation are selected at random and awarded prizes, with values ranging from ￡25 to ￡1,000,000. Overall, the annual interest rate is currently around 1.4%, but with this method most bond holders will receive 0%, while a few will win many times more than the actual bond value of ￡1, up to one million pounds.

So, your initial outlay is safe when you buy a premium bond – you can always cash them in at the price you paid for them – but you are gambling with the interest.

Now, the interesting thing from a statistical point of view is the monthly selection of the winning bonds. Each month there are nearly 3 million winning bonds, most of which win the minimum prize of ￡25, but 2 of which win the maximum of a million pounds. All these winning bonds have to be selected at random. But how?

As you probably know, the National Lottery is based on a single set of numbers that are randomly generated through the physical mechanism of the mixing and selection of numbered balls. But this method of random number generation is completely impractical for the random selection of several million winning bonds each month. So, a method of statistical simulation is required.

In a previous post we already discussed the idea of simulation in a statistical context. In fact, it turns out to be fairly straightforward to generate mathematically a series of numbers that, to all intents and purposes, look random. I’ll discuss this technique in a future post, but the basic idea is that there are certain formulae which, when used recursively, generate a sequence of numbers that are essentially indistinguishable from a series of random numbers.

But here’s the thing: the numbers are not really random at all. If you know the formula and the current value in the sequence, you can calculate exactly the next value in the sequence. And the next one. And so on.

Strictly, a sequence of numbers generated this way is called ‘pseudo-random’, which is a fancy way of saying ‘pretend-random’. They look random, but they’re not. For most statistical purposes, the difference between a sequence that looks random and is genuinely random is unimportant, so this method is used as the basis for simulation procedures. But for the random selection of Premium Bond winners, there are obvious logistic and moral problems in using a sequence of numbers that is actually predictable, even if it looks entirely random.

For this reason, Ernie was invented. Ernie is a random number generator. But to ensure the numbers are genuinely random, it incorporates a genuine physical process whose behaviour is entirely random. A mathematical representation of the state of this physical process then leads to the random numbers.

The very first Ernie is shown in the second picture above. It was first used in 1957, was the size of a van and used a gas neon diode to induce the randomness. Though effective, this version of Ernie was fairly slow, generating just 2000 numbers per hour. It was subsequently killed-off and replaced with ever-more efficient designs over the years.

The third picture above shows Ernie (mark 4), which has been in operation from 2004 up until this weekend. In place of gas diodes, it used thermal noise in transistors to generate the required randomness, which in turn generated the numbers. Clearly, in terms of size, this version was a big improvement on Ernie (mark 1), being about the size of a normal PC. It was also much more efficient, being able to generate one million numbers in an hour.

But Ernie (mark 4) is no more. The final picture above shows Ernie (mark 5), which came into operation this weekend, shown against the tip of a pencil. It’s essentially a microchip. And of course, the evolution of computing equipment the size of a van to the size of a pencil head over the last 60 years or so is a familiar story. Indeed Ernie (mark 5) is considerably faster – by a factor of 42.5 or so – even compared to Ernie (mark 4), despite the size reduction. But what really makes the new version of Ernie stand out is that the physical process that induces the randomness has fundamentally changed. One way or another, all the previous versions used thermal noise to generate the randomness; Ernie (mark 5) uses quantum random variation in light signals.

More information on the evolution of Ernie can be found here. A slightly more technical account of the way thermal noise was used to generate randomness in each of the Ernie’s up to mark 4 is given here. The basis of the quantum technology for Ernie mark 5 is that when a photon is emitted towards a semi-transparent surface, is either reflected or transmitted at random. Converting these outcomes into 0/1 bit values, forms the building block of random number generation.

Incidentally, although the randomness in the physical processes built into Ernie should guarantee that the numbers generated are random, checks on the output are carried out by the Government Actuary’s Department to ensure that the output can genuinely be regarded as random. In fact they apply four tests to the sequence:

- Frequency: do all digits occur (approximately) equally often?
- Serial: do all consecutive number pairs occur (approximately) equally often?
- Poker: do poker combinations (4 identical digits; 3 identical digits; two pairs; one pair; all different) occur as often as they should in consecutive numbers?
- Correlation: do pairs of digits at different spacings in bond numbers have approximately the correct correlation that would be expected under randomness?

In the 60 or so years that premium Bonds have been in circulation, the monthly numbers generated by each of the successive Ernie’s have never failed to pass these tests.

However:

Finally, in case you’re disappointed that I started this post with a gratuitous reference to Sesame Street which I didn’t follow-up on, here’s a link to 10 facts and statistics about Sesame Street.