View Full Version : Lottery probability question

07-31-2005, 08:20 PM
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 /images/graemlins/grin.gif 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,


08-01-2005, 10:35 AM
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.

08-01-2005, 02:07 PM
The average return on premium bonds is set by National Savings & Investments. I believe it is currently 3.25% per year. Of course, that's tax-free so it could be worth 4.06% to 5.42% to you depending on your tax bracket.

Your prize list comes to 72,750,000, which implies 26,861,538,462 of units outstanding, if my 3.25% figure is correct. By my calculation (which differs from the prior one because I'm not using 24,000,000,000 as the total outstanding), you need to wait 198 years to have a 50% chance of winning any prize 10,000 or larger.

There isn't really a lot of gambling with these bonds. Every 15 years, you expect to get one prize of 1,000 or more. During that time, you expect to get 43,875 total, almost all through the accumulation of 50 and 100 prizes. Even if the big prize is 1,000 or 5,000 or 10,000; it's not a huge difference over 15 years. There are 1,650 of those prizes, compared to the 37 that would really make a difference: 20 25,000; 10 50,000; 5 100,000 and 2 1,000,000. So about once every 15 years you get a 40 to 1 shot to make a difference.