Premium Bonds - Odds of winning

InMyDreams

masonic wrote: »

The figure posted is derived from a mathematical expectation, not a probability.

OK, now I'm confused... which figure? The only 'expected return' figure I see is the £31.70 in your post #3. I didn't use this figure and neither did MILLWALLFC. I think he multiplied 37.4% by 3 to get 112.2% which doesn't make any sense. According to your post #3 (which were the figures I used and didn't check), 37.4% was the probability of winning a £50 prize with a £10k holding. (Granted, I read that as 'at least' one £50 prize rather than 'exactly' one £50 prize.)

masonic wrote: »

Specifically, it is the probability of winning one £50 prize, plus (the probability of winning 2 £50 prizes) x 2, plus (the probability of winning 3 £50 prizes) x 3, plus...(the probability of winning n £50 prizes) x n, where n is the number of £50 prizes on offer.

Oh, yes, that's what I meant. So I understood that bit then. :T So that bit *is* a probability.

masonic wrote: »

All of the figures posted, including the ones in Martin's article are as well, since they are the sum of the probability of each bond winning a particular prize. I am using "expect" in the mathematical sense as well, not saying you will win £50 every month, which you wouldn't - on average you will win just over £50 a month in £50 prizes.

OK, not read Martin's article. Just using your figures. 'Are as well' what? As I said, the only 'expected' figure I saw was the £31.70 Had MILLWALLFC multiplied that figure by 3 it would have made more sense. You could expect to win 3*£31.70 (=~£95) each month with a £30k holding.

masonic wrote: »

Just to add, this would be ok, but the ~42.65% you have started with is actually the chance of winning with 1 bond (1 in 24,000 or 0.004166%) multiplied by 10000 [with some rounding errors, apparently]

Hmm... the ~42.65% I started with was from adding up *your* probabilities for "the chances of winning each prize, say, for a £10k holding per draw". Yes, I rounded the answer but I don't think that detracts from the final figures as I assumed there were probably rounding errors already with the figures I was starting with. I didn't use the 1 in 24,000 figure as you'd already worked out the probabilities for 10K so I didn't bother.

masonic wrote: »

so you have to start with one bond, then multiply up as you have done.

??? I assumed you'd already done that for your 10k figures. Didn't want to start from scratch again.

masonic wrote: »

So, that would make the probability of not winning anything with 1 bond 99.99583% (not surprising!), and the probability of not winning anything with £30,000 = 0.9999583^30000 = 28.6%. Incidentally, that makes the real figure for the probability of winning *anything* with £10k 34.08%, not 42.65%.

So are you now saying that...

At 24000 to 1 per bond of winning a prize, the chances of winning each prize, say, for a £10k holding per draw are:-

£1,000,000 0.000056%
£100,000 0.000846%
£50,000 0.00166%
£25,000 0.00333%
£10,000 0.00838%
£5,000 0.0167%
£1,000 0.190%
£500 0.570%
£100 3.46%
£50 37.4%
£0 34.08%

Umm... something doesn't add up here.

masonic wrote: »

Who says premium bonds aren't fun.

Yup

InMyDreams

InMyDreams wrote: »

masonic wrote: »

The figure posted is derived from a mathematical expectation, not a probability. Specifically, it is the probability of winning one £50 prize, plus (the probability of winning 2 £50 prizes) x 2, plus (the probability of winning 3 £50 prizes) x 3, plus...(the probability of winning n £50 prizes) x n, where n is the number of £50 prizes on offer.

Oh, yes, that's what I meant. So I understood that bit then. :T So that bit *is* a probability.

Oh, hang on, no it wasn't... how does the x2, x3 and so on come into it. I was reading this as 37.4% was the chance of winning at least one £50 prize but nothing bigger.

InMyDreams wrote: »

Umm... something doesn't add up here.

I guess that'll be down to my interpretation of the %ages you posted. I had assumed they were mutually exclusive and exhaustive probabilities (only excluding the 'don't win anything' outcome).

masonic

InMyDreams wrote: »

Oh, hang on, no it wasn't... how does the x2, x3 and so on come into it. I was reading this as 37.4% was the chance of winning at least one £50 prize but nothing bigger.

I guess that'll be down to my interpretation of the %ages you posted. I had assumed they were mutually exclusive and exhaustive probabilities (only excluding the 'don't win anything' outcome).

This seems to be where the confusion arises. They are not mutually exclusive probabilities. They are the matematical expectation, divided by the prize value. The '37.4%' is the expected income from £50 prizes, divided by £50. It is therefore the weighted sum of the probabilities of winning 1 prize, or 2 prizes, or 3 prizes etc (winning two prizes gives you double the winnings, hence the x2 bit etc.) That percentage therefore represents the number of £50 prizes you can expect (on average) to win each month with that holding. Maybe my expressing it as a percentage is why you are having issues with it.

InMyDreams

masonic wrote: »

This seems to be where the confusion arises. They are not mutually exclusive probabilities. They are the matematical expectation, divided by the prize value. The '37.4%' is the expected income from £50 prizes, divided by £50. It is therefore the weighted sum of the probabilities of winning 1 prize, or 2 prizes, or 3 prizes etc (winning two prizes gives you double the winnings, hence the x2 bit etc.) That percentage therefore represents the number of £50 prizes you can expect (on average) to win each month with that holding.

Oh, OK. So what is is a percentage of then? 37.4% of £50, ie £18.75 in £50 prizes? Yes, that makes more sense as an expected win. I was indeed confused by your terminology. You've got to admit, calling it a 'chance of winning' and then using a % rather than giving it in £ is a bit misleading ;-) That'll teach me to take short cuts and use someone-else's figures without checking them or what they mean

So in answer to MILLWALLFC, his 'chance of winning £50' is *not* 112.2% as he asked (which at best is meaningless and at worst implies certainty) but his expected win is 112.2% of £50 in £50 prizes. Ie £56.10 or just over 1 £50 prize per month as you later clarified.

Sorry to be pedantic. But it's this sort of ambiguity that sells PB in the first place, I think. You may argue that that's obviously what you meant, but I beg to differ. If I want to add 4, 5 and 6 together, I could write 4+5=9+6=15 and it's 'obvious' what I mean, but at the same time wrong and plain confusing to someone who didn't know what I was trying to do.

masonic wrote: »

Maybe my expressing it as a percentage is why you are having issues with it.

Nope, it was calling it a 'chance' in your first post that was what threw me.

masonic

InMyDreams wrote: »

Nope, it was calling it a 'chance' in your first post that was what threw me.

I hold my hands up, I can see that is confusing now. It looked like exactly what I was trying to convey at the time . I tried to explain to millwallfc what was meant by that figure, but maybe I wasn't clear enough there, either. By the way, there is no need to apologise for being pedantic - I don't want to leave anyone with the wrong idea about premium bonds any more than you do.