The maths behind this calculator is incredibly complex. It took three months to develop; and to do it I recruited a post doctoral researcher in cosmology and statistics to calculate the probability graph. Frankly even explaining how it works is difficult. Yet for maths nerds, here goes...
Using National Savings & Investments' (NS&I) official prize distribution information, the probability of each individual prize being won with one bond in one draw was easily calculated. Once these are known, the program employs a multinomial probability generating function to find the number of prizes any combination of bonds and monthly draws is likely to yield. From that, the likely cash value of those prizes, by looking at all possible scenarios, is found.
To crunch the numbers takes a powerful computer server over 12 hours and then the answers are placed in a database from which the information above is calculated.
How up-to-date is it?
The calculations are updated each time NS&I change the distribution of prizes (usually once a month), as it only publishes the distribution after the draw has been made, and unless stated don't take into account the irregular 'anniversary' draws which offer a couple more jackpots (though actually these have very little impact on the overall probability).
This may mean if NS&I has increased or decreased the interest rate since the last draw took place, we will list the old interest rate, as the rate can't be updated until a new draw is done.
Why when I enter a figure is it sometimes slightly different?
Due to the vastness of the sums needed, we decided to automatically round up or down some amounts of bonds when you plug them in. For instance, if you have £995 in Premium Bonds, the calculator gives you (and tells you it's doing it) the results for £1,000 worth.
We've deliberately chosen figures to round to that hardly affected the probability; it simply allows our databases to remain manageable. Doing every single value would mean our computers running for days each time it needs to update!
How accurate is the calculator?
With probabilities, there is always a slight fluffing as it's all about chance, as for every person carrying a four leaf clover, another has crossed a black cat while walking under a ladder. Yet if you take it as read that some will of course win a million, these results are very reflective of the reality for most Premium Bond holders.
There are a couple of mathematical assumptions made, purely to ease the burden on our website servers. We treat the selection of winning bonds in each monthly prize draw as independent events. This is an extremely good approximation as the number of bonds is so much larger than the number of prizes available.
Also, very very unlikely events are ignored in order to get the calculation done in a reasonable time, while maintaining the reliability of the results. The methodology used in the calculator has been verified and approved for us, separately to the algorithm designer, by a lecturer in applied financial mathematics at the London School of Economics.
