Saturday, 18 June 2011

How to win the lottery. Part 1: The Odds

I like to keep my posts, like my love-making, short and/or sweet, so when this one started ballooning out to Oracian, or possibly John Holmesian proportions, I decided to break it up into a trilogy of geeky maths delights, guaranteed to show you how to win the lottery.

I have bought precisely one lottery ticket in my life. I was fifteen and it was technically illegal. Luckily, I won jack shit.

I gave up, because you'd have to be stupid to buy a lottery ticket. Or so I keep telling myself.

Do you buy lottery tickets? If so, the following conversation may sound familiar to you, especially if your name is Bob.
"Hiya Bob, what brings you to the local newsagent where I buy my pornography."
"Hey Cedric, I'm just picking up my weekly quickpick."
"Tattslotto, eh? Did you know that the odds of winning Tattslotto are more than eight million to one against? Eight million!"
"I think I heard something like that. Still, you've got to be in it to win it."
"Uh uh. Sure, someone will probably win, but, statistically speaking, it is virtually impossible for you to win. You don't buy Powerball too, by any chance? "
"Er, sometimes, if there's a jackpot."
"Fifty five million to one*. Fifty five! Million! Man, you are such a Sucker. You won't catch me throwing my money away on this delusion. Those odds are seriously crazy. Did you know that the chance of getting hit by lightning is only one in 1.6 million? Know anyone who has been hit by lightning? Ha ha! Now, if you'll excuse me, I believe that the latest issue of Brazilians get brazilians is out." 
"Right then. See you later, Prick**." 
Cedric seems like a nice enough guy, and I like calculating big numbers and have as much interest in the goings-on behind the curtains at beauty salons as the next man, but the thing is, those odds are not true - or at least not honest. How could they be? It is a ruse invented by the anti-fun brigade. I mean, c'mon? Multiple people win these lotteries nearly every week.

Where then, do the odds above come from? Are they:
  1. The odds of any particular person winning the lottery?
  2. The chance of an average ticket winning the lottery? 
  3. The lifetime chance of winning for a regular lottery ticket purchaser?
  4. The odds that a single game will win? 
Of course it is the last one, not because it is in any way meaningful, but because it is the biggest number that can be concocted. The thing is, you can't even buy just one game when you play Tattslotto or Powerball. The minimum number of games you can purchase is four, and therefore the worst odds you can possibly get from a single Tattslotto ticket is actually two million to one, and Powerball likewise fourteen million to one. Most people who play the lottery buy multiple games and play regularly, often for many years.
"How about some realistic odds, Prick?"

I have invented a man - let's call him Mr Sucker. He buys a Tattslotto quickpick at the local newsagency every week for his adult life, which just happens to be exactly 50 years. What are his chances of winning? I can tell you upfront that they are a crapload better than 1 in 8 million, although, to be honest, they still kind of suck.

A regular quickpick is $7.85 and gives Mr Sucker twelve games. His chance of winning each week is therefore 12 in 8,145,060, or 1 in 678,755, for a payout of, on average, about $1 million. But that is just 1st division. If Mr Sucker gets five right and a supplementary, then he could win 2nd division and a cool $10,000, and so on down to 6th division (two numbers and a supplementary), where he basically gets his money back plus a scratchy for the wife. And he gets these same odds every week for 50 years.

Each of his tickets gives him the following approximate odds.
Division 1 - 1 in 679,000
Division 2 - 1 in 57,000
Division 3 - 1 in 3,000
Division 4 - 1 in 68
Division 5 - 1 in 25
Division 6 - 1 in 12
Each year, his odds of winning 1st division are around 1 in 13,000. I wouldn't bet my Millennium Falcon on it, but when you look at the odds this way, you start to get a different perspective on the chances of winning. Each year, he is more than 99% likely to win at least one prize, he has a two in three chance of winning division 5 ($20) and will probably win division 4 (about $30) every two years or so.

Over his lifetime, the odds of winning at least once are:
Division 1 - 1 in 262
Division 2 - 1 in 22
Division 3 - 1 in 0.7
Division 4 - 1 in 10-17
Division 5 - 1 in 10-47
Division 6 - 1 in 10-100
These are the real odds that matter. The ones that keep the punters coming back for more, week in week out. As you can see, Mr Sucker is odds on to win division 3 at least once, and is guaranteed to win many prizes. In fact, if we want to talk big numbers, the chance of him not winning at least one division 4 prize is 1 in 100 trillion, and the chance of him not winning division 6 at least once is 1 in a googol, and that is a damn big number. And on top of all that, he actually has a chance of winning first division. It's a small chance, but a real chance.

So maybe Mr Sucker isn't such a sucker after all. He will undoubtedly win many prizes, but, overall, how much money is he likely to win or lose? Well, over 50 years, we can actually say this with some confidence.

Part two: The Payout, coming soon.


Incidentally, the 1 in 1.6 million chance of being struck by lightning is, I think, actually the chance of being killed by lightning in any one year (about 10 people in Australia), although I couldn't find a good source for this. The lifetime chance is more like 1 in 20,000, while the chance in a weekly Tattslotto cycle is a whopping 1 in 83 million. What do you think of that, Prick?

*In the U.S., Powerball odds are 1 in 195 million.
** The conversation doesn't seem nearly as nasty when you realise that their names are Bob Sucker, and Cedric Prick.


Anonymous said...

The chances of winning any money in any lottery game are actually 50:50.
Either you will or you won't...

Anonymous said...

This post is wrong in two ways, although one is due to a change by Tattersalls.

1. The Powerball odds were changed in 2012 and are now some 80 million to one for first division.

2. The mythical man playing every week for 50 years is also wrong in giving reduced odds, and this is a common mistake. It was vigorously argued when I was doing high school statistics. The truth is that every game is independent and so your odds are always they same, they do NOT improve by repeat playing. This is a common, but wrong perception.

Daryl Holland said...

If you play for your whole life the odds of winning are much greater than if you play once.

E.g. If you roll a six sided dice 10 times you are much more likely to get a six than if you roll it once.

The odds of getting a six each time you roll don't improve, just like the odds of winning with each ticket don't improve. But overall your odds of winning with 10 rolls/tickets are much better than with just one.

I'm not suggesting the odds of the man winning with the last ticket he buys is 500 to 1, it isn't, it is 8 million to one. These are the odds of winning with any ticket over his whole lifetime, at the time he buys his first ticket.

Anonymous said...

Lotto draws are not completely independent because they are drawn under very similar conditions. Theory of Distribution states that outcome of similar events of equal probability will be distributed (roughly) evenly over possible outcomes given enough events. Casinos depend on this - this is why they always win because even if a punter has a good run the distribution will eventually come back in the casinos favour (games always have more outcomes in favour of the house).

And the 50/50 comment is just wrong. There are millions of outcomes that result in a "loss" but only 1 that results in the "win" (division 1 anyway).