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Lotteries: Mathmatically speaking, they’re not really a tax on the stupid

By RickMeasham | December 30, 2008

Note: There are a lot of numbers in this post, so I’ve rounded a lot of them off, and included some footnotes. If you don’t like numbers, then consider yourself warned.

It’s been said* that “Lotteries are a tax on the stupid”. The premise being that anyone who spends money on a lottery ticket is just wasting their money are, therefore, stupid.

I’d like to refute that.

Here in Australia, our Tattslotto lottery requires the player to select 6 numbers from 45 to win division one. That gives you one chance in 8.1 million1. This is approximately the same odds as tossing a coin and getting ‘heads’ 23 times in a row2.

So what other forms of gambling can we think of to compare it to? Roulette has one chance in 38 (or 37) of winning ‘the big one’3. Blackjack has approximately one chance in 41 of getting a ‘natural’ blackjack4. The chance of getting a royal flush in poker is approximately one in 650,0005.

But not all forms of gambling are equal. In some, the house pays you: In roulette, everyone who bet on that single number wins 35 times their wager no matter how many people are playing. On the other hand in poker, everyone who gets a royal-flush has to split the pot between them. Tattslotto is like poker in this sense. If 20 people all pick the correct 6 numbers, then you each get one-twentieth of the first division prize.

And of course, not all forms of gambling have the same payout. Roulette pays 35:1, but the other examples all vary. Poker varies depending on the number of players and how much they’re all betting. Tattslotto varies depending on how much the promoter wishes to pay out.

So if lotteries are truly a tax on the stupid, then surely the return on investment must be so truly microscopic that any sane person would laugh and walk away.

Next Saturday, Tattslotto has a $30 million ‘megadraw’. This means that first division is $8.4 million6. Each ‘game’ costs $0.60 and as we said earlier, each game has a 1 in 8.1 million chance of winning. To be ‘certain’ of winning would thus require 8.1 million games at a cost of $0.60 each. An investment of $4.86 million.

Now, if I took that same $4.86 million down to the casino and spread it among the 38 ‘number’ squares I’d also be ‘certain’ of winning. But I’d just win $4.47 million.

Now, let’s flip it around. What if we put our money on a single square in roulette? Our chance of winning is one in 38. It pays 35:1, so to win $8.4 million we’d need to bet $249K on the single square. In Tattslotto, to get a one-in-38 chance of winning, we’d need to play about 213,000 games, which would cost us $128K — about half the investment for the same risk and for the same reward in roulette!

Of course, I’m ignoring that ‘difference’: In roulette, it wouldn’t matter how many other people bet on the same winning number, I’d still get my $8.4 million. In Tattslotto, we all have to share the $8.4 million. It’s impossible to calculate the number of people that will pick the winning combination, so I’m going to ignore it for now except to say that if you shared it with one other person, your $4.86 million investment would get you a $4.2 million return — which is only slightly less than you get for the same investment in roulette. Can you truly call that a ‘tax on the stupid’? Depends on the individual’s definition of ’stupid’.

However, given all this, there’s one more argument against the ‘tax on the stupid’ theory. And it’s one that can’t be refuted by anyone: What are the odds of winning if you don’t play at all? By playing a single game, your odds of winning decrease from infinite to finite.

* NOTE: The best attribution I can find for this quote is a chapter title in James Walsh’s True Odds : How Risk Affects Your Everyday Life published in 1996. It the quote really that recent? If you have an older reference, please let me know.

Footnotes

  1. Odds of winning at Tattslotto are calculated by the number of balls that you’re happy to get divided by the number of balls available. For the first ball, you’re happy with any of 6 balls from a pool of 45. For the second ball, you’re happy with any of 5 balls (you already have one of your picks) from a pool of 44 (the ball that was drawn first isn’t re-entered, so you only have 44 left in the pool).This means that the chance of winning is one in (6 ÷ 45) × (5 ÷ 44) × (4 ÷ 43) × (3 ÷ 42) × (2 ÷ 41) × (1 ÷ 40) = 0.0000001228 = 1 ÷ 8145060.
  2. The odds of throwing a ‘head’ when tossing a coin is 1 in 2. So as above, we divide the results we’re ‘happy’ with (heads) by the results possible (heads or tails). That is 1 ÷ 2 = 0.5. If we flip twice then we have (1 ÷ 2) × (1 ÷ 2) = 0.25. If we flip 23 times we get 0.0000001192.
  3. There are two roulette wheels in common use: The European wheel has the numbers 1 to 36 and a ‘zero’ and the American wheel has 1 to 36, a zero and a ‘double zero’ (and recently, in some casinos in Australia). Despite the lengthened odds of the American wheel, the payout remains the same. So if you really have enough money to be putting $249K on a single square, fly to Europe first.
  4. To get a ‘natural 21′ in blackjack you need to draw an Ace and a 10 or picture card. As the order of drawing them doesn’t matter, there’s a 4 ÷ 52 chance of drawing an ace on the first card and a 16 ÷ 51 chance of drawing a ‘ten’ card on the second. The other way around and there’s a 16 ÷ 52 chance of drawing a ‘ten’ card on the first and a 4 ÷ 51 of drawing an ace on the second. No matter which way around that is, it’s the same (remember BODMAS, PEDMAS, BOMDAS or whatever your school called it?). Of course, if you’re actually playing Blackjack then there’s at least a dealer and that changes the odds.(4 ÷ 52) x (16 ÷ 51) = 0.0241327300 = 1 ÷ 41.44
  5. A royal flush is 10, J, Q, K, A of a single suit (hearts, diamonds, spades, clubs). The suit has no relevance other than it has to be the same. So for the first card, there are 5 happy cards in each suit. Each suit has 13 cards, so a successful first card is 5 ÷ 13. However after that first card, we do care about the suit: it has to be the same as the first card. So the second card can be one of the other four cards we need out of the 51 cards left in the deck: 4 ÷ 51.This means the chance of a royal flush is (5 ÷ 13) × (4 ÷ 51) × (3 ÷ 50) × (2 ÷ 49) × (1 ÷ 48) = 0.0000015391 or 1 ÷ 649740.

    Now, at this stage some of your are going to get itchy commenting fingers and want to point out that in poker, there’s at least one other person playing. This means there’s (4 ÷ 51) chance that he or she would get the second card you want and so the chance of getting your second card is (1 – (4 ÷ 51)) × (4 ÷ 50). What does this mean for the chance of you getting a royal flush? Bupkis. Nothing. Nada. Zilch. You might find out that you failed earlier, but they don’t change the odds of you getting a royal flush. Don’t believe me? Here’s a google-doc to show that while the distribution of the odds changes, the final result doesn’t. He takes a card, which might be yours, thus decreasing the chance of your flush. But he also takes a card that might not be yours, which increases the chance of your flush. (I’ll admit that I was surprised by the result, so I checked with the boffins in #math on the freenode IRC network and they agreed with me).

  6. This may surprise some of the folks who have seen the advertisements for the “$30 million megadraw” but they’re not talking about first division. They’re talking about the total prize pool. First division is 28% of the total prize pool.

Topics: Issues | 2 Comments »

2 Responses to “Lotteries: Mathmatically speaking, they’re not really a tax on the stupid”

  1. Tom Jewell Says:
    January 12th, 2009 at 7:26 pm

    But if they are offering $30 million megadraw then the chances are they are getting more than that in (off course mega draws may well be a bit different on that front). So it is quite likely a second person will also have the 6 numbers.

    You can consider if you are playing lotteries to try and increase your expected return. I.e Reduce the chances of how many people you share winning pool with. Certain numbers are likely to be chosen more often (dates of birth, first 6 numbers). Problem with this sort of technique is you don’t know how many other people are doing it.

  2. Smokey Says:
    March 26th, 2009 at 4:32 pm

    The $30m was for division 1 only. This from the Tattersals website:

    Division Prize Pool Winners
    1 $30,000,000.14 22 provisional Winners of $1,363,636.37 each.

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