One of the complaints that Maths teachers hear more than any other subject is “But we will never use that in real life.” Nobody ever says that about Shakespeare or quantum physics, which is strange.
So here’s an example where a bit of thinking about maths could save a motza in real life.
A number of online betting companies are currently offering huge prizes for correctly guessing the final make-up of the AFL ladder- that is getting all the teams in the right position at the end of the season. One ad that I saw is offering a prize of $100 million for a $5 bet. Wow! What a deal!
Hold on, let’s consider the odds. We will assume that at the beginning of the season all teams have an equal probability of standing in any position on the ladder. Although some teams are more likely to fill the top 4 or the bottom 4 you really have no way of knowing in advance the possible effects of injuries or even disciplinary action by the AFL over drug use or salary cap infringements.
There are 18 teams in the AFL. That means that the number of ways that the ladder can stand at the end of the season are:
18x17x16x15x14x13x12x11x10x9x8x5x4x3x2x1= 6400000000000000 (rounded off, but the odd digit here makes no difference). Remember those classes on combinations and permutations?
That means if you place one bet you have a 1 in 6400000000000000 chance of winning.
And the odds being offered are 100 million to 5 or 20 million (20000000) to 1.
So let’s knock off a few zeroes to find out what the margin is for the bookmaker on this competition: 640000000
0000000 to 2 0000000 or 320,000,000 to 1.
In other words the scale is stacked 320,000,000 times in favour of the bookie.
To put that in words:
- the odds of winning are astronomically remote 1 in 6.4 x10^15
- the payout is so ridiculously stacked in favour of the bookie that even if every combination is covered by a $5 bet, they pay out the $100 million but rake in 320 million x $5 or 1600 million
- to make sure that their payout is capped they limit the number of bets a single person can make and also stipulate that if more than one person picks the final result, they share the prize.
- Even if all 20 million (give or take) people in Australia take up the opportunity and put on the maximum number of bets (say 20 each) that is still only 400 million bets and the probability of the bookies paying out is still only 1 in 15 million.
That is why you need to pay attention to your maths teacher. The bookie did and that’s why he drives a Lexus when you drive an old Ford.