Premium Bonds Calculator Discussion Area

Former_MSE_Dan · 19 July 2007 at 10:53AM

This thread is specifically to discuss the
Premium Bond Probability Calculator itself

To discuss premium bonds themselves see the
Premium Bonds: Are they worth it? Discussion
linked from the
Premium Bonds: Are they worth it? Article

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ossian · 23 July 2007 at 1:00PM

This article appeared just as I was looking at Premium Bonds attracted by their tax position for higher rate tax payers and the ability to buy them in regular instalments unlike their index-linked savings certificates.

I've posted another thread as I am now completely puzzled on what to do with my £500/month that I intend to invest. Being a higher rate tax payer is a pain.

The_Bleurk · 23 July 2007 at 5:43PM

I read this piece in the Sunday Times with interest. I have no doubt that the clever maths may well be needed to create the database you have outlined but I think there is a far simpler method which requires only the distribution probability of winning a prize for a given stake and the actual average prize awarded - both obtainable from Ernie. Calculating the expectation value is then very straightforward.

Take my own case: I have a £30k stake which should yield 1.15 prizes a month. My own average prize over the last couple of years has been £60 which therefore gives 1.15*60*12= 828 for the year - or 4.6% equivalent for higher rate tax payer. In practice my prize rate has been 1.25 and that brings the higher rate equivalent up to 5%.

For the general case we need the actual average prize - I haven't had time to go and get that but it should be a simple calculation.

As to the odds of winning a million: surely if there is a total of £36bn staked then by simple division for a £30k stake I have 1 in 1.2m chance each month.

Or maybe I'm just lucky? :beer:

The_Bleurk · 23 July 2007 at 5:47PM

By the way....

Putting that last bit another way....

Statistically speaking I am certain to win a million once if I keep my £30k in there for 100000 years.

Maybe I should sup up...

The_Bleurk · 23 July 2007 at 6:18PM

OK I pushed myself - using the numbers available from Ernie's website for the prize distibution the forecast average prize for September is almost exactly £76 - so for a £30k stake the expectation value for the year is 1.15 * 76 *12 = £1048 or 3.5% or 5.83% for a higher rate taxpayer.

NB this is the expectation value - what you actually get won't equate to this in practice due to the scarcity of the higher prizes you will likely make a bit less than this unless of course you hit a big one.

In my book that means this is a worthwhile bet despite the impression given in the Sunday Times.

btw I forgot there are 2 prizes of £1m each month so that means I only have to wait 50000 years to be statistically sure I get my prize.

Aegis · 23 July 2007 at 6:27PM

The_Bleurk wrote: »

By the way....

Putting that last bit another way....

Statistically speaking I am certain to win a million once if I keep my £30k in there for 100000 years.

Maybe I should sup up...

Actually statistically speaking you're not. As the number of draws increases, you're more and more likely to win, but it's actually an exponential curve that asymptotically approaches a probability of 1 as time goes on.

The reason that this is the case can be modelled with, say, dice. You have a 1 in 6 chance of rolling a 6 each time, which means you have a 5/6 chance of not rolling a 6. The chance of rolling 2 dice and getting no 6 is (5/6)^2, and this extends to the general case of:

P(No 6 in N dice) = (5/6)^N

Or, to rephrase:

P(1 or more 6s in N dice) = 1 - (5/6)^N

Now, the Limit of (5/6)^N as N tends to infinity is 1, but the value itself is never actually 1 for any finite value of N. As such, there is a slim chance that even if you rolled billions of dice you might not roll a single 6. A very small chance, but a chance anyway.

For the specific situation you mentioned, the probability boils down to:

P(winning the big prize if playing for N months) = 1 - ((1-1/1.2E6))^N

So if N = 1200000 then we have:

P = 63%

So, playing for 100000 years only gets you a 63% chance of winning the jackpot if your figures are right!

Bonbon · 23 July 2007 at 6:33PM

why doesn't the calculator give me a chance of winning the million? I thought every number has an equal chance of winning.

Bonbon · 23 July 2007 at 6:34PM

maths isn't my best point and now it's gone over my head

nrsql · 23 July 2007 at 6:53PM

The_Bleurk wrote: »

Statistically speaking I am certain to win a million once if I keep my £30k in there for 100000 years.

Nope - You are never certain to win a million or in fact any prize.
Each draw is independent so you have the same chance of winning in each draw - it makes no difference if you haven't won anything in the previous 100000 years.
To calculate probabilities in this sort of scenario multiply the probabilities of not winning and subtract from 1 - obviously this can never result in 1 (certainty). It becomes increasingly les likely to not win as you consider more draws - but the probability for the next draw doesn't change.

beingjdc · 23 July 2007 at 6:59PM

Bonbon wrote: »

why doesn't the calculator give me a chance of winning the million? I thought every number has an equal chance of winning.

It does, it's just too small to show up.

The_Bleurk · 23 July 2007 at 9:22PM

maybe I should have said:

on average I should win a million once in 50000 years...

In any event it seems like a good bet to me given that I get my stake back whenever I want. I would like to retrieve it in 50000 years...

Not too much to ask?

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