Betting against the system – Part 1

I was assured that there is no question of failure. There is no set of circumstances that could prevent the fulfilment of his prediction. The assurance I was given was that by 25 December 2010, my work colleague will win the lottery.

This display of psychic foretelling naturally piqued my interest. It seems my colleague was inspired by the 34-year-old Limpopo man who, this past November, won a R30 million jackpot in the first Powerball lottery draw, after previously winning a National lottery jackpot of R11 million in 2002. This colleague (let’s call him Lot) had stepped back from his original claim, uttered just minutes before, that he would win twice in 2010 to his new claim that he would win once (after all, why should he be greedy?) and it would be R10 million, enough to ensure a comfortable life well into retirement.

To explain why I wagered my money and to explore some of the logical fallacies in which gamblers engage, I will present the arguments in these three parts:

  • Part 1 – A little mathematics can be dangerous.
  • Part 2 – It doesn’t take much to convince yourself.
  • Part 3 – Is it a winning system?

Part 1 – A little mathematics can be dangerous

Having done just a little mathematics at University, and some statistics for good measure, I have long understood that it isn’t wise to plan your future around winning the lottery. My attitude to the lottery (and casinos too, by the way) is captured by the phrase; “The lottery is a tax on people who don’t understand mathematics.” But how could I start the discussion without being dismissed as a nay-sayer? A number of ordinary challenges crossed my mind: “impossible”; “never”; “are you barking mad?” Finally I settled on one that would at least get the debate going; “If you win the lottery jackpot in 2010, I will pay you R10 000.”

I didn’t intend any kind of trickery when I made this offer, it is in force at this moment. If Lot wins the lottery’s top prize (of any amount, not specifically R10 million, because it varies from week to week) between 1 January 2010 and 25 December 2010 (these happen to be dates of his choosing) then he will get a nice cash bonus from me. He can play as many tickets per week as he wants to and the tickets can be for the National Lottery, the Lotto-plus draw or the Powerball lottery, any of the six draws per week for 51 weeks. If I had the money, I would have offered R100 000 or R1 million – it’s not that I didn’t offer those amounts in case “things go bad” but rather that it wouldn’t be a sincere offer if I wasn’t able to have the promised cash available at the moment of the win. It’s worth noting that there’s nothing in this agreement for me, I get nothing in return if Lot doesn’t win, the winning criteria have been relaxed from his original claim to further improve his chances.

How can I offer the money and be so sure that he won’t win? Isn’t this just a case of opposite beliefs where it’s Lot’s unshakable belief in his certain win versus my unshakable belief in his certain loss? No, I am not 100% certain that he won’t win, there is a tiny possibility that I will be dishing-out 10 grand in the new year (I understand this fact but I bank on how tiny that possibility really is). In other words; I am doing this because I know the odds are very firmly on my side – a well-understood fact, so much so that another colleague of mine matched my offer to hand-over R10 000 to the future millionaire in our midst. Contrast my attitude, having weighed the odds, to Lot’s attitude; he is ignoring the odds, hoping for a win.

When pressed for justification for his certainty, Lot presented a string of logical fallacies that I will discuss and he revealed that he has a system. I will share with you the details of the system (that he chose to share with me) after a little diversion into why regular gamblers draw conclusions based on bad thinking.

“I just can’t lose”

Imagine your child has a school fund-raiser (for some of you this will first require imagining a child – bear with me, it won’t be long). The school would like to build a new library, say, and they decide to sell raffle tickets each week until they have the money to build that library (we can assume they go on selling tickets forever, because the cost of construction increases with inflation making the target just out of reach). For a small fee of R100 you can buy one (and only one) of two hundred uniquely-numbered tickets (let’s assume all of them are always sold out) and, at the end of the week, one of the ticket numbers will be chosen to win R6 000 by a process, required by the Laws of the Universe to involve a large noisy wire-cage ping pong ball draw machine and a draw organiser who must complain briefly and then fumble for a pair of glasses to delay the announcement of the winning number.

It’s a simple scenario, you have your one ticket out of two hundred possible tickets. And by the way, you are free to stick with the same number each week or change whenever you like, it’s up to you and it doesn’t change the maths at all. So based on this simple scenario, here is the question; how many weeks do you have to play to be guaranteed a win? Take a moment to think about it. Really do, it’s easy to keep reading to see the answer, but try to figure it out. If you arrive at a number, try to find a calculation that results in an answer that agrees with your estimate.

Perhaps, as I did some years ago, you decided that two hundred attempts would guarantee a win. No, I wasn’t even close. Perhaps your reasoning was similar to mine; playing for one week your chance is 1 in 200, playing for a second week is another 1 in 200 chance, adding these chances together you see that playing for two weeks results in a 2 in 200 chance of winning. Continue forward and you reach a 200 in 200 chance (or dead certainty) after 200 weeks. That simply isn’t true. The logic of adding your chances together is only valid if each chance is in the same draw – i.e. if you buy 200 tickets in one draw, you are guaranteed to win. But remember that by the rules of our game you can only buy one ticket per week.

Rephrasing our ticket’s odds of winning allows us to find the real answer. Try thinking about it this way; instead of saying your odds of winning are 1 in 200, rather say that you are guaranteed a win except in 199 of 200 outcomes. Let’s write that as follows
\frac{1} {200}  \: = \:  1 - ( \frac{199} {200} )

This second form allows us to do something pretty nifty, we can represent the shrinking odds against us that each new week introduces by multiplication, after two draws, our odds are: 1 - ( \frac{199} {200} \cdot  \frac{199} {200} ) \: = \: 0,009975 \: ( \text{ or 0,998} \%)
and after the third week the odds are: 1 - ( \frac{199} {200} \cdot \frac{199} {200} \cdot \frac{199} {200} ) \approx 0,01493 \: ( \text{ or 1,49} \% )

This pattern gives us the general formula:
1 \: - \: ( \; 1 - \frac{1} {p} \; )^n
where “p” is the total number of raffle tickets (not your tickets, the pool of all of the tickets) and “n” is the number of draws that you participate in. Let’s plug the values for our school raffle scenario into the formula and see if it is true that after 200 draws we are guaranteed a R6 000 payout:
1 \; - \; ( \; 1 \; - \; \frac{1} {200} \; )^{200} \; \approx \; 0,633042178274 \: ( \text{ or about 63,3} \% )

That means playing 200 times only gives me about a 63,3% chance of winning. Wait a minute! I paid R20 000 to get a 63,3% chance of winning R6 000? Okay, but seriously, that must be a fluke because surely there must be a point where we are guaranteed to win? It has to happen right? What about playing 500 times (rewriting your child’s homework assignments should keep him or her in primary school for long enough to participate), surely then I’m guaranteed to have won at least once? No, the probability of winning at least once increases to 91,84% but it still isn’t 100%. So what is the magic number? How many times do you have to play to get 100%? The answer is an infinite number of times. Analysing a huge number of weeks reveals to us the exponential trend, for example; 10 000 weeks (more than 192 years) of playing a 1 in 200 raffle still results in a probability of about 99,9999999999999999999829859597% There is still that tiny possibility that your number will never come-up, and by this point you’ve forked-out up to R1 000 000 in order to try and win R6 000.

Graph of y = 1 - ( 1 - 1÷200 )x

Incidentally, when “n” = “p” and as the value of “n” increases it approaches a constant value which we can use as a shortcut whenever we want to calculate the probability (and it’s far fewer buttons to press on the calculator).

By definition:
\lim_{n \to +\infty} \: \: 1 \; - \; ( 1 - \frac {1} {n} ) ^n  \: \; = \: \; 1 - e^{-1}

I’m sure you’ll agree that 1 – e-1 is far easier to enter into a calculator. This value is approximately 0,632120558829 (compare that to the value calculated when “n” = “p” = 200 earlier and try some larger numbers like “n” = “p” = 13 946).

Let me clarify one point before I continue; I’m not saying that it’s impossible to win, that would be absurd, it could happen that you win in the first week, it could also happen that you win twice in one year and it could happen that you win each and every time you play, but these are scenarios which have decreasing probabilities to the point of almost zero probability in the last scenario. What I am saying is that, despite the simple odds in our school raffle there is no fool-proof system that will guarantee a win.

In a raffle, somebody has to win, it’s designed that way; all participants have a number which nobody else can have and one of the numbers will be picked. Even for a system as simple as a raffle, there is no way to “guarantee” a win, somebody must win but the likelihood that the particular winning-somebody is you is not guaranteed. Consider that the next time you hear someone say; “I just can’t lose.”

“Somebody must win”

Although it is true by definition that a raffle must have only one winner, lotteries are vastly different to raffles and it takes a little mind shift to see that the odds of being the one lottery winner are far worse than being the one raffle winner. Imagine a simple lottery where participants only have to choose one number from 1 to 200, even in a small group of participants some people could possibly pick the same number (not knowing someone else picked that number) and the winners would have to share the winnings. On the other hand, if millions of people played the same game and they didn’t know the numbers that other people picked, there is also the small possibility that nobody picked a certain number. Because of this there is also a possibility that the winning number was not picked by anyone even when there are more players than the winning odds. When people talk about the lottery and say that “somebody must win” it simply isn’t true. In our example, the chance of you winning the lottery is 1 in 200, but the chance that you are the only winner is smaller and depends on the number of players.

“Picking six numbers less than 50, the odds aren’t bad at all”

Let’s talk about roulette for a moment because there is an illustration in this game which works well when considering the odds in a lottery draw. If I was to ask you to bet R30 on just one number in a roulette spin, you might do so quite easily. Your payment of R30 gives you a 1 in 74 chance of winning R2 160, not a lot of money compared to the lottery but it’s still a gamble; think of all the 73 possibilities which end in failure. Now let’s assume you won but you are not allowed to take your winnings, all of that money must stay on the same number for another spin. You would naturally be very nervous because the chances in the second spin are still 73 in 74 that you’ll leave with nothing. Risky, but if you did win, you’d have R155 520, no amount to be sneezed at if you only wagered R30 (just imagine if it was R35 or R45).

© WikiMedia, Connor Ogle

But I’m a cruel games-master, yet another time you are not allowed to take the money, you are forced to spin a third time. This time if you win you cash-in R11 197 440 in chips, more than double the average lottery payout for the top prize. But would you play such a game? If you won twice in a row, would you be tempted to spin again or take 155K? I have painted the only one scenario in which you win R11 million, but there are 405 223 scenarios in which you leave with nothing! It’s easy to fixate on the one wining scenario, but the odds are overwhelmingly against this scenario happening.

If you spoke to a friend, and that friend boasted that they could win at roulette by betting three times on the same number and winning each of those three times, would you hesitate to say they were unlikely to win in this scenario? Would you bet against them? After all, the chances of them winning the bet is 1 : 405 224 but your odds of winning the bet are 405 223 : 405 224 (or 99,99975328%).

But we’re talking about the lottery here not roulette, so what are the odds of winning the lottery prize? We’ll find the answer in a branch of mathematics called Combinatorics. I won’t go into all the details of the calculation (which you can find here) but the short answer is that the total number of combinations of six unique drawn numbers ranging from 1 to 49 is C(49, 6) = 13 983 816. The odds are then 1 : 13 983 816 (it’s worth noting that the possible combinations in the Powerball draw are 20 × C(49, 5) = 38 137 680 which means it’s more than twice as likely that you have a winning National Lottery ticket than a winning Powerball ticket). The odds 1 : 13 983 816, don’t appear to be so bad, and the flawed thinking goes that it’s far less than the population of the country and how many different people have you seen around? It’s easy, with our brains which are unaccustomed to large numbers, to mistake this for being a relatively small number, but when was the last time you saw 14 million of anything and you could see each and every one of the set? If you’re thinking about a full stadium at rugby or cricket matches then you’re far too short at about fifty~ to seventy thousand spectators. Even the largest sport spectatorship is 250 000 at the Indianapolis speedway, picturing such large numbers is difficult but we are still short of our target.

When you compare our lottery odds (1 : 13 983 816) to the odds of winning three-spins-in-a-row at the roulette table (1 : 405 224) we see that it is FAR MORE likely that you would win the roulette triple-spin game 31 TIMES before winning the lottery even once.

The risks we take every day

Every day we (South Africans) risk our lives just by living, by doing nothing extraordinary, and yet these odds haven’t yet come up for us:

  • 1 : 2 386 chance you will be the driver of a vehicle involved in a fatal accident this year
  • 1 : 3 647 chance you will have a fatal heart attack this year
  • 1 : 2 248 chance you will have a fatal stroke this year
  • 1 : 3 000 chance that you will be struck by lightning at some time in your life
  • and on any single international airline trip, there is a 1 : 543 026 chance you won’t arrive at the destination

And despite the lottery being less likely than all of these, we still think it is inevitable we will win the lottery and live long enough to see that day.

As a reward for reading all the way through this post, I have given you all the winning numbers for the lottery draw for the second week in February 2010, the winning numbers are all in this post. Remember you must be 18 to play and winners know when to stop. 😉

~ by James on 27 December 2009.

5 Responses to “Betting against the system – Part 1”

  1. Great post James. I enjoyed that… Oh. And Angela bought a couple of lottery tickets the other day. Feel free to make fun of her. 🙂

  2. […] was James’ explanation of an encounter he had with a colleague who has been fooled by the national lottery. James clearly explains the maths behind why winning the lotto is highly unlikely and gives clear […]

  3. James, thanks for a cool and interesting post. In terms of the perception of chance and our (humans’) hard-wired difficulties in evaluating risk vs gain, are you familiar with the work of Bruce Schneier? I was struck by the similarity in basic ideas between your description of the difficulty of evaluating what large numbers mean — vis-a-vis odds of hitting the lottery relative to “daily” catastrophes — and his discussion of cognitive bias in evaluating risk. (For example: While he approaches the topic from a security-oriented perspective, I see some underlying similarities in his approach to the subject that you’re considering.

    Again, thanks for an educating, approachable and entertaining post.

    • Thanks for the interest SJR. Yes, I am very familiar with Schneier – I am a software developer and have been fascinated by security for many years. I don’t recall reading the particular article that you linked to but I have come across these concepts repeated by many authors from different perspectives. I think that there’s a kind of predictability in people (whether it the common misunderstanding of risk versus reward – such as the Monty Hall Problem – or whether it’s our lack of a good mental PRNG) and understanding people is crucial for security.

  4. Sorry, I do not speak English. (
    Recently he became interested in gambling on the Internet! It is very interesting, even started a blog! Thanks for the great post, a lot of useful find for yourself!)

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