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# The mathematics of spot-fixing

## It is human nature to want to gamble. But that desire is founded on probabilities

Spot-fixing: suddenly on a whole lot of minds. Three young cricketers accused of doing it for no real reason I can fathom except greed. After all, they were already earning money legitimately far in excess of the great majority of their countrymen.

Still, I’m not here to pass judgement on these men. They are innocent until we find otherwise, and that finding will eventually come from a court. And anyway, who knows what motivates young men with lots of money? No, I’m here to discuss what makes spot-fixing possible; especially, some of the mathematics behind it all.

But let’s start with this: what makes a bet possible? Of course, I suspect it is almost human nature to want to gamble. But that desire is founded on probabilities. You consider an upcoming event, you estimate the probability of it turning out a certain way, and you choose to place a bet (or not) based on that estimate. There are fellows called bookies who will take your bet. Based on their own estimate of what’s going to happen, they will give you what’s called “odds" on the event.

For example: Imagine two cricket captains about to toss a coin. Both of them, and all of us, know the probability of it landing heads is 1/2. If you find a bookie willing to take a bet on this, it’s likely he’ll give you odds of 1:1; meaning, for every rupee you bet, you’ll get a rupee back if the coin does in fact land heads. A pretty stupid bet to make, you’ll agree. Because if you keep betting, you’ll lose your rupee half the time—when the coin lands tails. And when it lands heads, you simply get your rupee back.

But consider tossing a dice instead. The probability of a “1" is 1/6, and that opens up more apparently interesting betting possibilities. A bookie will likely offer odds of 5:1 on a “1"; that is, for every rupee you bet, you’ll get back five if the dice shows “1". (If it shows anything else, you lose your rupee.) Sounds exciting, this chance to quintuple (wow!) your money? Would you take these odds and place a bet like this?

Yet here’s the thing, and this is why I used the word “apparently" above. Please don’t stop breathing at the mention of quintupling your money. For the mathematics says this is actually just as stupid a bet to make as with the coin. Again, if you keep betting, you’ll lose your rupee five out of every six times. (Put it another way: five of every six bettors who place such a bet will lose their money.) Only once—that sixth time—will you get your five-rupee windfall.

The reason bookies might offer such odds—1:1 for the coin, 5:1 for the dice—is that they know their probabilities as well as you do, and naturally they don’t want to lose money. In fact, they will likely tweak the odds they offer just enough so they actually make money. That is, after all, why they do what they do.

So if you find a bookie offering quite different odds than you expect, it’s likely he knows something you don’t. Consider how that might pan out. Let’s say the coin the captains use is actually a fake—it has tails on both sides. But let’s say only our devious bookie knows this. He says to you the avid bettor: “Ten times your money back if it comes up heads!" You think: “Wow! There’s an attractive proposition!" and you gamble 1,000, for you’ve estimated that there’s a 50-50 chance you’re going to waltz home with 10,000.

Then you lose, as—face it—you were always likely to do. Bookie laughs all the way to the bank with your 1,000.

All of which is essentially how spot-fixing must work.

So now imagine you are a fervent cricket-watcher. (Which I’m willing to bet you are, unless you’re Lady Gaga.) From years following the game, you know that bowler J bowls a no-ball about once in every six-ball over. Along comes bookie W to whisper in your ear: “Psst! Hundred times your money back if J bowls exactly one no-ball in his first over in the Siliguri Master Chefs game!" Your eyes widen and you fork out the 10,000 you didn’t win when he offered you the coin bet, starry visions of a million-rupee payoff whirling through your head. Hundreds of other cricket fanatics like you do the same. (Rather silly cricket fanatics, but never mind.)

What you don’t know, of course, is that bookie W has instructed bowler J to bowl not just one, but two no-balls in that first over. For doing so, J will get a slice of all the money W has collected in bets.

So J bowls his two no-balls at the Master Chefs. You lose. Bookie W and bowler J laugh all the way to the bank. Simple.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. A Matter of Numbers will explore the joy of mathematics, with occasional forays into other sciences. To read Dilip D’Souza’s previous columns, go to www.livemint.com/dilipdsouza

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