Lottery probability question

Hi, I was discussing this with a friend and we got stuck fairly quickly. If anyone could find the time to do some calculation I'd really appreciate it.

There's a sort of "lottery" (Premium Bonds) in the UK whereby you buy £1 units in a prize draw and each £1 you've invested gets one shot a month at winning a variety of cash prizes. You don't lose your investment if you fail to win, and you can get it back at any time.

Let's say for the sake of argument you can buy a maximum of 90,000 units.

The odds of one unit winning a prize in the monthly draw is 24,000-to-1 against. (90k ~ 3.75 prizes a month)

The prizes are distributed as follows (rounded):



6% of prize pool -

2 x 1,000,000 units
5 x 100,000
10 x 50,000
20 x 25,000
50 x 10,000
100 x 5,000



5% of prize pool -

1500 x 1000 units
4500 x 500



89% of prize pool -

180,000 x 100 units
930,000 x 50 units



What we were trying to calculate is how long you'd have to hold 90k units to be almost certain (95%) of winning the million. We got stuck because we're stupid [img]/images/graemlins/grin.gif[/img] However I'd now like to know the following (probably phrased badly because I'm not exactly sure what I'm getting at):



1) How many months would you have to hold 90,000 units to have a 95% chance of hitting one of the top 3 prizes?

2) How many months would you have to hold 90,000 units to have a 95% chance of winning a prize in the top band?

3) What is the average monthly return of 90,000 units? Like, 3.75%? Or am I missing something obvious?

4) What is the probability of winning at least 100,000 units after 6 months with 90k?



I hope this is enough information to solve the problem. We got stuck and I'm not sure whether the problem is that we don't know how many other people have invested the maximum or whether that affects your odds of winning (I have no common sense whatsoever).

I guess it's sort of an expected value problem. My friend says that buying 90,000 is -EV if it means you're only going to get a small return after a few months while your 90k is tied up with no utility. If anyone can think of any other questions regarding the profitability and suitable timescale of investing the maximum I'd be very interested.

I hope I don't come across as stupid; I'm just reading my first poker book now (HoH) and trying out a few EV and probability calculations but I'd like to get an answer from someone knowledgeable.

Thanks in advance,

James