In ireland the goverment issue a special "Savings Bond" called Prize bonds. You dont get any interest on them, but you can get your money back any time you want, and you are put into a weekly draw with a chance to win various money prizes as follows:
1 x €20,000 x 52 - 12 (12 prizes of €1M) = 40 prizes per year of €20K
5 x € 1,000 x 52 = 260 prizes per year of €1K
10 x € 250 x 52 = 520 prizes per year of €250
Over 7,400 x € 75 x 52 = 384,800 prizes per year of €75
Currently there are 7,400 prizes in each draw. So given the figures, currently there is 7400 – (10 + 5 + 1) prizes each week. (origionally this was 7000) Lets ignore this minor calculation as additional prizes are added as the fund grows.
Also there is 1 monthly jackpot prize of €1,000,000, which replaces the €20,000 prize in that week. As the prize bond fund grows they add additional prizes to the €75 category. Even if you win, no matter the prize, you can keep winning. So your tickets are not removed once you win.
On their www.prizebonds.ie site they say a customer who has €1,000 invested has a 3.6 to 1 chance of winning. But I dont get how they work this out. Or even what it means. 3.6 to 1 (3.6 of what to 1 of what). If its 3.6 chances to win for each bond then every bond would be winning 3.6 times in a year. This doesnt seem real. But I am open to being wrong. I just dont really get this.
To purchase a prize bond, you get them in lots of 4 at a time, each €6.25 each or €25 in total. This entitles to you 4 entries in each draw.
In 2010 the amount of money in sales of prize bonds was €1,328,000,000. From this figure we can obtain the number of investors. €1,328,000,000 divided by €6.25 each equals 212,480,000 prize bonds have been issued.
My question is two fold. How do I work out my chances of winning if I had invested €1000 and leave it there for the period of the full year either 31-dec-09 or 1st-jan-10, to 31-dec-09 or 1st-jan-11 (I know maths guys are picky so was not sure exactly what dates to use to represent the full year, lets not get into leap years :-> ).
Simple as this may seem, when I calculate this, I dont get 3.6 to 1 as they do in their FAQ. So this is the second part of the question. How are they getting this figure. And what exactly does 3.6 to 1 mean. The part of the FAQ that I refer to is following:
I would prefer my chances of winning expressed as a percentage. For example, I get X number of bonds for my €1,000 euro, each prize will appear X number of times over the 12 months, and there are X number of prize bonds compteting with mine for those prizes. This makes sense to me. But how do I convert my answers to something like 3.6 to 1.
Thanks in advance for your help.
Edits for consideration
The edits have been highlighted in bold. If anyone is woundering where the previous answers got their figures I want new people to know that above the origional figures were €398,000,000. These were the number of sales in 2010. As at least one answer pointed out, I should have used the number of actual bonds outstanding as opposed to the number sold. So I changed that number above. The figures are got from the following link:
When there scroll down to the second heading under "PRIZE BONDS SALES GROW BY", there is a short paragraph which shows figures. The one I am interested in is "Fund Value at Year End", which has a figure of €1,328,000,000 (1.328 billion).
This can be used to tell us that at the end of 2010 exactly €1,328,000,000 divided by 6.25 equals 212,480,000 (origionally I had 63,680,000).
So I am updating those figures above.
Also, I am a bit of a mathematical idiot. So I dont understand most of the first 3 answers I have received. What is that squiggly sign that looks like an equals having cramps. Is there an easy way to explain this stuff. I hope this is not offensive. But I dont have strong mathematical background. Thanks again.
Best Answer
Regarding odds, Wikipedia has a clear explanation stating that usually odds-against rather than odds-on are stated, and that m to n odds corresponds to success probability of n/(m+n). So, odds stated as "3.6 to 1 chance of winning a cash prize in a 12 month period" corresponds to likelihood of $1/4.6 \approx 21.7\%$.
In computing the odds yourself, two problems arise. First, the 2010 sales you mentioned (€398,000,000) may be much less than the actual total capitalization, against which your €1,000 investment is competing. (I didn't find any sales or cap. figures for prizebonds, so will just continue with the numbers you gave.) Second, last month's payout at prizebonds was ca. €3.9M, about 65% more than the ca. €2.4M payout one would expect from the table of draws you mentioned (from prizebonds faq #11).
The first problem will make the odds worse; the second, better. All that said, I think getting within a factor of two or so on the odds may be as good as one can expect. Here's one such calculation:
Probability of success on one draw = investment shares / cap. shares = investment / cap. = 1000/398000000 $\approx$ .0000025.
Probability of 1 success in a year = Binomial dist.(1; n, p) where n = draws per year, so nominally n $=52\cdot7016=364832$, and p = .0000025 from above.
Now $n\cdot p \approx 0.96 < 10$ so a Poisson approximation with $\lambda = 0.96$ is ok. From a Poisson article, Prob($k;\lambda)$ = $\lambda^k\cdot e^{-\lambda}\over{k!}$ whence Prob(1; 0.96) = $s \approx 0.37$, or 37% chance of a payment within 12 months time. Hence, the odds based on this calculation are $(1-s)/s = 0.63/0.37 =$ 1.7 : 1, vs. official odds of 3.6 to 1.
Update 1 Although I was aware Prob($1;\lambda)$ isn't the desired answer, I assumed (incorrectly) that Prob($k>1;\lambda)$ would be of negligible magnitude. However, $\text{Prob}(k>1;\lambda)\approx 0.237$. $\text{Prob}(k\ge1;\lambda) = 1-\text{Prob}(0;\lambda)\approx 0.60015$, giving odds of 2 to 3.