An interesting proposition just came up on my twitter feed. Tom Waterhouse, the smug git you see on TV spruiking his bookmaking service, is offering $25 million dollars to anyone who can pick the first 10 runners in the Melbourne Cup in order - and it only costs you $10 per try! (apparently you get 10 tries at most, don't get greedy now!)
Sounds good, right?
Well, no. Intuitively most people can smell a rat straight away - after all, it's difficult to win the lottery and that's only 6 numbers that need picking (though there are more of them). But it's far worse than that. Even with two horses scratched, if we naively assume each horse has the same chance of winning, then picking the first horse is a 1 in 22 chance. Picking the next horse is then a 1 in 21 chance (as we can eliminate the winner), then so on until the 10th horse is a 1 in 13 chance. All up, the probability of doing this is about 1 in 2 million million. Even if the entire population of Australia at about 23.6 million (including children!) put in their 10 bets each, there would be an 0.01% chance that anyone would win.
The smart gamblers are probably now thinking "well, each horse doesn't have an equal chance of winning - I can exploit that!". And they'd be right! So let's look at the odds of each horse winning (I've used the fixed odds at Betfair but feel free to substitute your own). We can estimate the probability of each horse winning from the bookmaker's odds, and this is as accurate a representation as we're likely to find without significant effort - after all, it's in the bookmaker's interests to know the probabilities as accurately as they can to make the most amount of money! A formula to estimate the probability of a particular horse winning is:
1/<the odds for your horse> divided by the sum of (1/odds) for every horse.
Doing this gives us a probability of about 16% of the favourite, Admire Rakti, winning. So let's put our money on the favourite winning, followed by the second favourite in second place, and so on. Once our favourite goes past the line, we then need the second favourite (either Fawkner or Lucia Valentina) to come next. We can estimate the probability of this happening by dividing its probability of winning (about 13%) by the probability of all the remaining horses' probabilities of winning combined (about 84%), giving us a probability of about 15% of this horse coming second given our favourite has already come first.
Again, we rinse and repeat until we get through the first ten horses. This gives us a much nicer final probability of 1 in 28 million. Again, naively you might think that if everyone in Australia had a go at this, surely with 236 million bets, we'd be able to do it pretty easily.
Unfortunately though, if everyone in Australia put their bets in here, they're not all going to be able to pick this most likely scenario. If they did then either everyone would win, sharing the $25 million dollars and getting $1 each from their outlay of $100, or nobody would win! Instead, everyone would have to organise to pick the 23.6 million best odds. And then, even if someone managed to win, Tom Waterhouse would still be pocketing $2,360 million dollars and only having to shell out $25 million, making his smug face even more unbearable...