[BritNewbie]

OK, let me have a stab at the first part of your question.

Suppose you hold just one bond.

Probability of winning a prize in any given month is 1 in 24001

Of the 1,116,187 prizes each month, only 8 are top 3 prizes; (ie. £1 000 000, £100 000, or £50 000)

So probability of your one bond winning a top 3 prize in any given month is

1/24001 x 8/1 116 187

= 2.986233 x 10^-10


Multiplying this by 90 000 units gives the probability of your 90 000 bonds winning a top 3 prize in any given month as

2.6876097 x 10^-5

So probability of your 90 000 units NOT winning a top 3 prize in any given month

= 1 - 2.6876097 x 10^-5

= 0.999973123903

Let n be the number of months that you would need to hold your bonds in order to be 95% certain of winning a top 3 prize.

Then n satisfies the following inequality:

0.999973123903 ^n < 0.05

By my reckoning, n needs to be greater than 111 464.

That is a tad over 9288 years.

You may like to know that the maximum allowed holding is actually 30 000 bonds, each costing £1. (It was 20 000 for a long time, but recently they increased it.)

If you held the maximum of 30 000 bonds, I reckon - using the above method - you'd have to hold 'em for something like 27 866 years to be 95% certain of winning a 'top 3' prize.

As in poker, so too in Premium Bonds, it seems - it's all about the long term.