May(un)fair? Your chances of winning at McDonald's Monopoly

Monopoly
McDonald's are once again running their Monopoly promotion, where along with your food you can collect stickers featuring properties from the famous board game. It's not just a matter of filling up a sticker book, of course - collecting properties can win you big prizes - but only if you manage to build up a full set of any particular colour. It's basically a lottery with a cunning twist - it's easy to feel you're closer to winning than you really are.
 
For example, to win the top prize of £500,000 you just have to find the two dark blue properties (Park Lane and Mayfair), but whilst there are plenty of the former going around, it seems, from reading the competition rules,that there is just one Mayfair sticker in the entire country.If this is true, it means that thousands of people will find themselves "just one sticker away" from winning a small fortune, when in reality their minuscule chance of winning has barely increased.
 
Naturally, I was interested to know what the odds are of finding that elusive Mayfair sticker - there must be millions of property stickers out there after all, but how many millions? I contacted McDonald's to see if they'd tell me but they were strangely unhelpful, so I had to find another route. Fortunately I can turn to another part of the promotion - the 'instant win' game. When you buy a product, you might get lucky and find that instead of a property sticker you find an 'instant win' sticker that entitles you to some particular prize immediately, no pesky property collection required. The total number of instant win stickers are detailed in the competition's rules, and comes to a whopping (that's Burger King, I know) 46,646,012. Knowing this, if I can get a random sample of stickers I can see what proportion of them are instant wins and from this estimate the total number of stickers in the game.
 
Unfortunately, I don't have a particularly strong interest in eating at McDonald's for every meal, so I turned to friends and colleagues to help me in this data collection exercise. At time of writing we've collected 89 stickers between us, of which just 14 (or 16%) were instant wins. This suggests that there are a little over 295 million stickers in total. If our estimate is correct, then any one sticker has a 1 in 295 million chance of being that £500,000 winning Mayfair. This is a staggeringly low probability - you have considerably more chance of winning the lottery. Twice.
 
That said, to be a proper statistician I should address a fairly important aspect of any attempt to estimate something from a sample - uncertainty. Just because 16% of my sample of stickers were instant wins doesn't mean that 16% of all stickers are instant wins, it's just the best estimate we can make from the data available. If you tossed a coin 10 times and got 6 heads, would you be happy to say that the 'true' chance of getting a head from that coin was exactly 60%? Probably not, but you might feel ok saying it's unlikely to be under 25% or over 95%, and the more times you tossed the coin the more sure you'd become about the true value, and the narrower this range would get.
 
We can use statistics to calculate such a range more formally, and this is known as a confidence interval. They are notoriously tricky beasts to define precisely, but a reasonable interpretation is that they offer a plausible range in which the value of what we're trying to estimate might lie. For my Monopoly data we get a confidence interval* for the total number of stickers of 195 million to 613 million. This is a pretty big range, but we can still see that (at least based on my dataset) that your chances of finding Mayfair are looking pretty awful.
 
So if you stumble across a Park Lane sticker with your next Big Mac and fries, don't get excited and start hunting for the other half of that set. Your waistline (and your wallet) will thank you in the long run.