Premium Bond Shenanigans

tom188

the first was that I think that the odds are adjusted to favour fresh entrants and complainers, as has been borne out by friends and family, you get most wins within three months then the tap is turned off.

As I said above this is complete rubbish. It is just as likely to happen then as any other time.

ManAtHome

Not my experience that you win more in the early days either - haven't bought any for around 4 years, but this year has been much better than the previous 3. Just the way it goes I suppose - you' re pretty much guarateed to win a million every 2,000 years or so (don't write in, that's just an illustration...), it could just as easily be the 1st month as the 91st or 23,000th...

Milarky

tom188 wrote:

As I said above this is complete rubbish. It is just as likely to happen then as any other time.

It's not complete rubbish - there is a basis to this belief.

In the early draws the element of 'randomness' is maximal. After two, three, four draws (etc) this randomness reduces rapidly. In the long run you experience very close to average numbers of prizes (we're not talking prize value here - only the numbers of prizes themselves!) In other words you can expect to say how many prizes you will win in (say) two years (24 draws) with high levels of confidence. If that average is one every month (about right on 25K) then you should have about 24 plus or minus a couple. You might have won none (or three or five) in that time, granted, but the chances of the numbers being 'significantly' (in all senses) different from '24' are slim. Statisticians call the range of prizes expected the confidence interval and the percentage of falling within that range, the level of confidence (be it 95% or 99%)

So going back the OP's tactic: by exposing yourself to relatively few draws you are maixmizing the element of randomness. Under such circumstances you have a greater likelihood of winnings 'away from the average' (numbers of prizes) and so the 'prediction' is that you will either win several fewer prizes (as seems to have happened) or several greater number of prizes...

tom188

In the early draws the element of 'randomness' is maximal. After two, three, four draws (etc) this randomness reduces rapidly. In the long run you experience very close to average numbers of prizes (we're not talking prize value here - only the numbers of prizes themselves!) In other words you can expect to say how many prizes you will win in (say) two years (24 draws) with high levels of confidence. If that average is one every month (about right on 25K) then you should have about 24 plus or minus a couple. You might have won none (or three or five) in that time, granted, but the chances of the numbers being 'significantly' (in all senses) different from '24' are slim. Statisticians call the range of prizes expected the confidence interval and the percentage of falling within that range, the level of confidence (be it 95% or 99%)

How can you have "maximal randomness". Something is either random or it isnt.

The confidence intervals of the parent distribution, where the observations come from, are independent of time. They will not change (other than due to anomalies caused by fluctuations of prize fund etc). It is just that as you continue to hold the bonds your sample distribution will be a better representation of the parent distribution.

Your statistical argument is flawed. Over a long time scale the mean number of prizes (pa?) you expect to win will converge to a typical value. However the probability of you winning a prize will remain unchanged. The probabilities do not change with time.

Thus as I said

This is complete rubbish. It is just as likely to happen then as any other time.

nrsql

Milarky wrote:

It's not complete rubbish - there is a basis to this belief.

In the early draws the element of 'randomness' is maximal. After two, three, four draws (etc) this randomness reduces rapidly. In the long run you experience very close to average numbers of prizes (we're not talking prize value here - only the numbers of prizes themselves!) In other words you can expect to say how many prizes you will win in (say) two years (24 draws) with high levels of confidence. If that average is one every month (about right on 25K) then you should have about 24 plus or minus a couple. You might have won none (or three or five) in that time, granted, but the chances of the numbers being 'significantly' (in all senses) different from '24' are slim. Statisticians call the range of prizes expected the confidence interval and the percentage of falling within that range, the level of confidence (be it 95% or 99%)

So going back the OP's tactic: by exposing yourself to relatively few draws you are maixmizing the element of randomness. Under such circumstances you have a greater likelihood of winnings 'away from the average' (numbers of prizes) and so the 'prediction' is that you will either win several fewer prizes (as seems to have happened) or several greater number of prizes...

This is a joke?
Think you might be trying to say the greater the sample the closer to the expected reuslt i.e. the more draws the nearer to the expected number of prizes..

Milarky

nrsql wrote:

This is a joke?

No joke, just reasoning...

nrsql wrote:

Think you might be trying to say the greater the sample the closer to the expected reuslt i.e. the more draws the nearer to the expected number of prizes..

Exactly, but by converse reasoning - the fewer the draws the, further what will be observed is likely to be from the 'expected' (i.e long run) distribution.

Suppose the odds of winning a prize are exacly 50:50 (and all prizes are of equal value):

After 1 draw possibilities are:

1 prize 50%
0 prize 50%

('mean' = 0.5)
In this 'extreme' case there is actually a 100% probability that the number of prizes won are not the mean (1/2 prize) and a 50% chance of this number being 100% (extra 1/2 prize) greater than the mean...

After two draws:

2 prizes 25%
1 prize 50%
0 prize 25%

('mean' = 1.0)
Here there is a 50% chance of hitting the 'mean' (so the mean is becoming more significant) whilst there is a reduced (albeit significant) 25% chance of the number of prizes won being 100% (1 extra prize in 2 draws) greater than the mean...

After three draws:

3 prizes 12.5%
2 prizes 37.5%
1 prize 37.5%
0 prize 12.5%

('mean' = 1.5)

Again there is a problem with it not being possible to win the mean number of prizes. But the one to watch is the 12.5% chance of winning +100% in prize money..

The pattern used is the familiar one from Pascals Trianagle:



It gives the number of 'chances' [in each box] that a number of prizes [position of box from the left] will be awarded.

Just looking at odd numbered rows is better for us - because in these a box lies in the median position. Also we must ignore the very first row because the first draw corresponds to the possibilites '[1][1]'.Thus in the fourth draw (fifith row) [1][4][6][4][1] we see there are six chances of winning 2 prizes, 4 each of winning 1 or 3 prizes and 1 each of 4 or 0 prizes. The total number of chances must add to 100% and will be 2^4 of 16.

After four draws:

4 prizes 1/16 = 6.25%
3 prizes 4/16 = 25%
2 prizes 6/16 = 37.5%
1 prize 4/16 = 25%
0 prize 1/16 = 6.25%

('mean' = 2.0)

So the possibilties are: 'mean score' [2]: 37.5%; 50% better than mean score [3]: 25%; 100% better [4]: 6.25%

See what has happened in a fairly short time (number of draws): There is still a chance of doing 100% better than the mean but it it is halved for each draw. Also there is now an intermediate position of gain (mean + 50%) and this is still significant - at 25%

The confidence intervals of the parent distribution, where the observations come from, are independent of time. They will not change (other than due to anomalies caused by fluctuations of prize fund etc). It is just that as you continue to hold the bonds your sample distribution will be a better representation of the parent distribution.

True but the individual player's distribution is not the parent distribution and never can be - it simply converges to the parent distribution.

HTH

tom188

Nonsense

Exactly, but by converse reasoning - the fewer the draws the, further what will be observed is likely to be from the 'expected' (i.e long run) distribution.

The results of the draw are just as likely to be far from the expected result now as at any other time.

Let n be each unit of bonds you hold.
After 1 draw you have n chances of winning x prizes
After 2 draws you have n chances of winning x prizes
After 3 draws you have n chances of winning x prizes
After 4 draws you have n chances of winning x prizes
After 5 draws you have n chances of winning x prizes

This is binomially distributed as you have said - You hold n units of which each has probability of winning a prize p.
The distribution does not change with time
Therfore the probability of winning x prizes does not change with time.

Therefore you are just as likely to win a prize at the beginning as at any other time. Which i believe is what Milarky is arguing is not the case.

These draws are independent events, therefore what has happened has no bearing on what will happen. You cannot therefore use the results of past draws or your record of successes to predict what will happen in the future. You are just as likely to win a prize after a month as after a year as after a century (presuming your holding is not diluted in this time).

The distribution of total prizes awarded to an individual over a shorter time period will be a less reliable estimate of the average winnings. However in no way will this influence the outcome of a particular draw. As it stands you are just as likely to win a prize in a draw at the when you first buy the bonds as at any other time.

Milarky

tom188 wrote:

The distribution does not change with time
Therfore the probability of winning x prizes does not change with time.

It does change with time. It changes each time there is another draw. This corresponds to moving done a row in Pascal's triangle - with a new set of coefficients. As I explained, the 'parent' distribution is simply what the long-term player's (or 'entrant' or whatever we call them) own distribution will converge to - given an infinite amount of time! He doesn't have an inifinite amount of time so we use a 'binomial type' (i.e. discrete) distribution instead.

To reiterate: The player's own distribution is discrete. Initially it is downright 'lumpy'. The distribution converges (quite rapidly) to the assumptions of normality (a 'continuous' distribution) but in the initial period there is a much higher likelihood that their winnings (expressed as a percentage return) will be significantly long or short of what those winnings can be expected to show over a number of years.

tom188

Do you agree with

At the 1 draw you have n chances of winning x prizes
At the 2 draw you have n chances of winning x prizes
At the 3 draw you have n chances of winning x prizes
At the 4 draw you have n chances of winning x prizes
At the 5 draw you have n chances of winning x prizes
...
At the t draw you have n chances of winning x prizes
(maybe I should have been more explicit with my original post).

Therefore the probability of winning a prize in a particular draw will not change in time
Therefore the number of prizes that you win in a particular draw is indepentent of time.

To reiterate: The player's own distribution is discrete. Initially it is downright 'lumpy'. The distribution converges (quite rapidly) to the assumptions of normality (a 'continuous' distribution) but in the initial period there is a much higher likelihood that their winnings (expressed as a percentage return) will be significantly long or short of what those winnings can be expected to show over a number of years.

Yes but my point is that the probability of you winning a prize in an early draw is just as large as after 100 draws. These are discrete random events, that occur independent of what has happened in the past.

Looking at your argument:
Say you were to start monitoring the results of your PBs after the 50th draw your bonds had been entered into. You would get your same results

eg
After 50 draws (1st one monitored) possibilities are:

1 prize 50%
0 prize 50%

After 51 draws (2nd one monitored) possibilities are:

2 prizes 25%
1 prize 50%
0 prize 25%

After 52 draws (3rd one monitored) possibilities are:

3 prizes 12.5%
2 prizes 37.5%
1 prize 37.5%
0 prize 12.5%

After 53 draws (4th one monitored) possibilities are:

4 prizes 1/16 = 6.25%
3 prizes 4/16 = 25%
2 prizes 6/16 = 37.5%
1 prize 4/16 = 25%
0 prize 1/16 = 6.25%

This pattern is just a consequence of increasing the period over which you monitor the draw - obviously the more draws you look at the closer the distribution will tend to the normal. However the time you start studying these data is irrelevant. Hence it makes no sense to say that you are more likely to win initially than after 50 draws.

GingerSte

I would be more inclined to agree with:

At the 1st draw you have n chances of winning x prizes
At the 2nd draw you have n chances of winning x prizes
At the 3rd draw you have n chances of winning x prizes
At the 4th draw you have n chances of winning x prizes
At the 5th draw you have n chances of winning x prizes

Hope you don't mind me sticking my oar in!