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How To Bet the Lotteries

In order to graduate from the Summerfield School of Business at the University of Kansas I was required to take and pass Calculus 101, Calculus 102, a course called Probability and Matrices, a course called Quantitative Analysis, and the dreaded Business 368, Statistics.  Those five courses in mathematics have thoroughly prepared me to dispense some 100%-authoritative wisdom about how to bet the lottery.  Here it is.

The best way to bet the lottery is by pretend.

If you do, and if you keep track, you will eventually become convinced that you shouldn't bet the lottery for real at all.

Why?  Because you simply cannot guess the winning numbers often enough.

Do the below sound familiar?

  • "If I do win, the payoff will be a lot more than what I've spent."
  • "Someone has to win."

If you use these rationales or any others to explain to yourself why you spend time and money buying lottery tickets, you need to keep reading.

Do the below sound familiar?

  • "I've been playing the same combination for six years now.  It's bound to hit soon."
  • "I always play the numbers my horoscope says to."
  • Or how about even this: "The combination 1-2-3-4-5 has hit six times out of the last six, so I'm going to bet that combination this time."

If you use these rationales or any others to help you decide which combination of numbers to choose, you need to keep reading.

Let's discuss first the sheer amount of time you spend choosing which lottery numbers to play.

Those little numbered balls don't know anything and they can't remember anything.  They're styrofoam (Styrofoam™, actually), and styrofoam has been scientifically proven to be completely non-viable and otherwise utterly incapable of memorizing anything and then recalling it.  Those little balls don't know which numbers came up last time or the time before that or any other time, and even if they did they still can't somehow force themselves into certain locations inside that whirlwind inside that drum.  It's completely random, and completely unpredictable, every single time.

You're not going to believe this, but the winning numbers for the last six days were the same!  Every single drawing for the last six days was the same, which is highly unlikely, no doubt.  Yes, every single drawing for the last six days was 1-2-3-4-5.  So what is the likelihood the seventh drawing will also be 1-2-3-4-5, given that it hit for the last six days in a row?

Answer: Exactly the same as for any other combination.

In a fair lottery your odds of winning with a particular number -- no matter what it is -- depend entirely and only on how many numbers there are to choose from, and that's all there is to it.

For a simplified example, if you have to guess five numbers in order in order to win, and if every number can range from 1 to 100, then the likelihood you'll win can be precisely calculated using this reasoning:

The probability of your guessing the first number is 1 in 100, or 1%.  The probability of guessing the second number is the same, .01, and so on for all five numbers.  Because this is not a conditional probability, all we need to do is multiply the probabilities of each independent event.  The formula for your calculator is .015, which means .01 raised to the power of 5, or .01 X .01 X .01 X .01 X .01.  The answer is .0000000001.

The likelihood you'll win, then, no matter what numbers you play, is exactly one in ten billion!  Said another way, if you play ten billion times you can expect to hit exactly once.  If you have a spare ten billion dollars (and a spare 318.88 years in which to make an average of one bet per second) you'll win once.  And if you do win, you'd better demand a lot more than ten billion dollars!  A correspondent tongue-in-cheekedly points out that if the same numbers came up six times in a row, he would assume the lottery is not fair and so he would bet those numbers on the seventh drawing.  In case you're wondering, the probability that the same given numbers will be chosen seven times in a row in a fair lottery is (.01 raised to the power of 5) raised to the power of 7.

The denominator of that number is a 1 followed by 70 zeroes, or 10 duovigintillion.  (Hover your cursor over that word to see it written out.)  If you like thinking about large numbers, go here, then don't forget to come back.

If you think that betting your "birthday numbers," whatever they are, on your birthday will confer some special luck on you, you're making the mistake of assuming that those little balls of styrofoam somehow care about you.  They don't.  They don't even know it's your birthday.  In fact, on any given day it's the birthday of well over 16 million people, so what makes you special?  And tomorrow it will be another 16 million people's birthdays.  As I say, those balls are styrofoam, they're petroleum popcorn, and their movements are entirely at the whim of the wind, not whatever day happens to be your birthday.

If you think that betting the same number over and over again in the hope that "it has to come up sooner or later" is a smart idea, you're forgetting that even the very brightest and most experienced of styrofoam balls can't remember what numbers have hit in the past.  (This is one version of what's called the "gambler's fallacy.")

The odds of any particular number hitting -- or not hitting, for that matter -- are exactly the same each time.  Repeat that to yourself every time you spend any extra time deciding which numbers to play, because any extra few seconds you spend deciding which numbers to play are wasted.  You might as well play the same numbers every time, or not, or anything else, because it just doesn't matter.

Also, if you think that tracking all the winning numbers and betting those is a good idea, you too have forgotten that styrofoam forgets.  (This is the other version of the gambler's fallacy.)

Here's a quiz: Which of these is the best bet?

  • The numbers that won yesterday
  • The numbers you played yesterday
  • The numbers 1-2-3-4-5
  • The numbers 1-2-3-4-6
  • The numbers 39-38-37-36-31
  • Your birthday numbers (on your birthday)

Yep, they're all equally likely, all equally unlikely.

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Now, as to the bad odds.

You certainly don't need to have taken any undergraduate-level math courses to understand that the payoff matrix in lotteries is weighted against you.  For example, if you bet a dollar and there are 1,000 possible choices then you should demand that the payoff for winning be $1,000, which means in the long run you'll break even, i.e., on average you'll win once in every 1,000 bets.  And even that is a poor use of your capital, because it only breaks you even.  Because of the time value of money, there's an opportunity cost even to breaking even, because you could have invested in an FDIC-insured savings account at some small rate of interest.

But lotteries don't work that way.  In a lottery, if the odds of winning are 1 out of 1,000, then the payoff might be only $500, which means in the long run if you place 1,000 one-dollar bets, which will cost you $1,000, you'll end up winning back only $500.

Imagine that I walked up to you on the street and made you this offer: "Let's flip a coin 1,000 times, and every time it comes up heads you have to pay me a dollar, but I'll pay you a dollar only every other time it comes up tails."  If you have more brains than money you'll turn me down flat, yet that scenario is no different from the lottery scenario described above.

Here's a related situation that always amazes me: Casinos frequently advertise the payoff ratio of their slot machines, saying something like,

Loosest slots in town!   97% payoff guaranteed!

What they're doing is admitting that you'll lose money on their slot machines.  They're specifically saying,

  • You give us a dollar and we'll give you back 97 cents.
  • We'll be happy to do that all day and all night, till you don't have any cents left.
  • And we hope you never figure out what a bad deal this is for you and what a good deal this is for us.
  • Oh, and thanks for being a moron.

If you chip in a measly dollar a month into an investment that pays a measly 5% compounded monthly for ten years, you'll have earned $35.38 in interest.  If you chip in a dollar a month to 97% slot machines, you'll have lost $3.60, a total difference of nearly $40.  Now multiply that difference by ten or a hundred dollars a month.  Then multiply it by several million people a year.  Casino owners are the smart people, and they're taking money from the stupid ones at a rate of several billions of dollars a year.

So, if you agree that a payout ratio of 97 cents on the dollar is a bad deal, consider the payout ratio of, say, the Kansas Lottery.  According to an official Kansas Lottery pamphlet issued in January of 2000, the prize money paid out was 53.75% of revenues.  Can you believe that?  That means that if you bet $100 over the course of, say, a year, you can expect to receive only $54 in payments.  You're out a whopping $46 annually, not to mention the lost interest revenue, for no good reason except ignorance of the laws of chance and the payout ratio.

HERE'S another example, from Minnesota's Mega Millions lottery.  As you can see if you read the very fine print, "The approximate overall odds of winning a Mega Millions prize are 1 in 14.7."  Yep, you read that right.

Let me make that really plain, because Minnesota doesn't mind doing so.  The state is telling you in no uncertain terms that if you give them $100 they'll probably give you back $14.70.

If you think this is a good deal, then I hereby officially and permanently offer you a better one.  I will pay you $1 every time you pay me only $25, not the $26 Minnesota wants.

If the Kansas lottery pays out only 54% of your investment, this exact bet pays an astonishingly low 3.8%.

(Also, despite what the lottery officials said, calculating the odds is not at all difficult.)

That slot machine's payout ratio of 97% is starting to look pretty good after all compared to the lotteries'.

If after all this you insist on playing the lotteries, you can maximize your payoff by following these two tips:

  • Choose cash, not an annuity.

By choosing cash rather than the annuity, you have more flexibility in your investments.  An annuity as offered by the lottery is a good deal only under two circumstances:

  • One is if prices are falling, i.e., we're seeing deflation, not inflation.  When prices are rising over time, an annuity pays back all but the first payment at dollars that are worth less, but when prices are falling, those later payments are worth more and more at the time they're paid.  Note that deflation rarely occurs; usually there's mild to rampant inflation, which is exactly what the lottery people are literally banking on.  Note also that if we do start to see deflation, you can be sure the lotteries will stop offering annuities.
  • The other circumstance is if you have so little self-control that you wouldn't be able to prevent yourself from blowing too much of your winnings if they were paid all at once.
  • Choose a number that's less likely to be chosen.

By choosing a number that's less likely to be chosen by the other bettors, you split the winnings -- in the unlikely event they accrue to you -- with fewer people.  As an example, on 9/9/99 you can be certain that a mass of mental midgets played many numbers with many nines.  Since the winning numbers had no relation to how many nines there are in the number you chose, you should have chosen one with no nines in it.

Anyway, the best way to bet the lottery is not to.  But if you must, always choose a number with a six in it, because six is a lucky number.

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An almost whole 'nother topic.  Let's say that, like in The Deer Hunter, someone loads a live round into a six-shot revolver and forces you to spin the chamber, aim it at your own head, and pull the trigger.   The question is, what is the likelihood you'll be shot if you're forced to keep spinning and shooting as many as three times?

If you're thinking, "Six chambers, three attempts, must be 50%," then think again.

The following is a rant against bad math, and it's not really all that funny, so be warned.

The easiest way to calculate your chances is to start by calculating the likelihood, the probability, that you'll be alive after the first trigger-pull.   That's easy:  It's 5/6.  The probability, expressed by mathematicians as p, that you'll survive the second attempt is 5/6 times 5/6.  The p of surviving three such attempts is 5/6 times 5/6 times 5/6, or ((5/6) ^ 3), which works out to 58%.

Therefore the p of being shot must be 1 (which is identical to 100%) minus 58%.  The 50% chance you might have originally thought is fully 19% larger than the actual answer of only 42%.  Still, don't try this at home.

Yet another topic.  Did you notice how 50% is 19% greater than 42%, not the 8% you might have thought if all you did was subtract 42% from 50%?  Way too many people get confused when comparing one percentage to another, and this is a good example.

When a percentage changes from 42% to 50%, that is indeed an increase of 8 percentage points, but it's not an increase of 8 percent.  Calculating a change in percentages is no different from calculating a change in butter or guns or anything else that's expressible as a number.

The rule for calculating a percentage change is this:

(New minus old) divided by old

Memorize that simple formula and you'll never again be stumped by how to calculate -- expressed in terms of a number with a percentage point -- a change from one number of somethings to a different number of somethings.

In brief, what the formula means is to divide the difference between the two numbers by the former (the one that's earlier in time) of those two numbers.  Said in yet more specific words, first you subtract the older number from the newer one, then you divide that difference by the older of the two numbers.

Increases.  Let's look at a couple of examples of increases:

  • Your butter inventory goes from 10 pounds to 15: Here the newer number is 15.  The older number, the one we're coming from, is 10.  The difference is 5 pounds.  5 pounds divided by 10 pounds is 50%, which comports with your instinctive understanding that 15 is half again bigger than 10.  So far so good.
  • What about a change from 8 guns to 10 guns?  Is that a 20% increase?  It's harder to see without employing the formula above.  The older value is 8 guns, the newer value is 10 guns, so the formula produces (10 - 8) divided by 8, which is not 20% but 25%.

    Now do you see why 25% is 25% greater than 20%, not merely 5% greater?

( .25 - .20 ) divided by .20 = 25%

Increases greater than 100%.  The formula also works for increases that are equal to or greater than 100%.  Here are two examples.

  • Your previous computer had 64 megs of RAM, and your new one has 128.   128 minus 64 is 64; and 64 divided by 64 results in an increase of 100%.

    Note that an increase of 100% means doubling.
  • Because of changes a few months ago in the zoning laws of your municipality, your supply of giraffes rises from 3 to 21. The percentage change is 18 divided by 3, or 600%. 

    Note that an increase of 600% means multiplying by 7, not by 6.  Similarly, an increase of 325% means multiplying by 4.25.

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percent_more.jpg (11,556 bytes) 07292003

The scan above is the first paragraph of a full-page, $12,000 ad paid for by BlueCross BlueShield of Kansas City in the July 29, 2003, edition of The Kansas City Star.

If, as the text says, Missouri surgeons pay 275% more than $31,500, how much do they pay in dollars?  Simply multiply $31,500 by 3.75, and you get the answer of $118,125, not the $87,000 alleged.

They meant, of course, 175% more.

Lesson: Don't believe eveything you read.

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Decreases.  The formula -- remember, it's (New minus old) divided by old -- works for decreases as well as for increases.  Here are two examples.

  • Let's say you're 5.5 years old and you tripped and fell and injured yourself 10 times last month, but this month you did it only 9 times.  What is the percentage change, in this case a decrease from 10 to 9?  Is it 10%?

    The difference is 9 - 10, or minus 1.  The older value is the 10 falls from last month, not the 9 from this month.  Minus 1 divided by 10 is  -10%, meaning a 10% decrease in your trip-and-fall injuries.  Good for you.

  • Let's just say, for no particular reason, that you bought shares of Panaco Oil (NASDAQ: PANA) at $4 and as of today it's selling for a whopping 60 cents.  What is the percentage change?  The newer value is .6, and the older value is the original 4.  That difference of a negative $3.40 divided by the older value of $4, then, is -85%.

    As in the example immediately above, if the answer is negative then that means a decrease, so in this case the answer of -85% means a loss in value of 85%.  Bad for you.

The special case of zero.  In all the examples above the two numbers were greater than zero: 15 pounds of butter, 8 guns, 12 giraffes.   What about the situation where one of the numbers is zero?  There are two possibilities, one in which the former number is zero and the other in which the latter number is zero.  These are special cases, because zero is a special number.

  • In the case of the newer number being zero, it doesn't matter what the older number is, because if you reduce a positive number of things -- whether they're guns or butter or dollars -- to zero then the percentage decrease is always 100%.  If you go from 12 giraffes to 0 giraffes, you don't need to implement the formula to see that's a 100% decrease.  Same thing if you start with a million dollars and go to 0 dollars, that's still a 100% decrease.

    Update of December 8, 2005: In a nationally broadcast advertisement on the FX television network this evening, AOL declared unequivocally, "Spyware slows browsing by up to 500%."  It would be interesting to ask AOL just what that means.  Once you've decreased browsing speed by 100% it has dropped to zero, so how can it drop by more than that?

  • In the case of the older number being zero -- in which, say, the number of giraffes is zero one month and 6 the next -- the formula above does fail because of nonsensicalness.

    If the former number is zero, as in the case of going from zero giraffes to even one, then the formula requires you to divide by zero, which doesn't make sense.  It's not that going from zero giraffes to 6 is a 600% increase, or even a 700% increase.  If it is anything, it is an increase of an infinitely large amount, but that doesn't matter.  What matters is that any formula can fail if it requires dividing any number by zero.  The only conclusion is that if a number rises from zero to any positive number, you simply cannot express in terms of percentage what an increase that is.

Negative numbers.  So far we've been dealing with cases in which both the old and the new numbers were non-negative.  Now we come to the remaining possibility, the seemingly more vexing problem of how to deal with negative numbers.

For example, it might not be immediately obvious what the percentage change is when you go from 10 giraffes to -2.5 giraffes.  But never fear: The formula still works exactly as advertised.  Now, I realize it's hard to imagine a negative number of giraffes, and it's almost as hard to imagine half a giraffe, so we'll switch to the example of dollars of profit or loss for two periods in a row.

Here are examples of the four possible cases:

  • An increase, with both numbers negative, e.g., from -100 to -25.  This is going from a loss of $100 last month to a loss of only $25 this month.
  • An increase from a negative number to a positive number, from -100 to 25, or from a loss of $100 to a profit of $25.
  • A decrease, with both numbers negative, e.g., from a loss of $25 to a loss of $100.
  • A decrease from a positive number to a negative number, e.g., from a profit of $25 to a loss of $100, which is the same as going from 25 to -100.

As I say, the formula still works.  The problem with negative numbers is only in interpreting the result.  In all the examples that didn't involve negative numbers, it was obvious whether the result of the formula was an increase or a decrease, and it was obvious what the term of expression was.

The term of expression is merely what unit of measurement we're using.  For example, in the giraffe problems above, the term of expression was simply giraffes.  But in these examples involving a negative number we have two terms of expression to consider, those being profit and loss. 

Profit is expressed as a positive number, as you would expect, and loss is expressed as a negative number.  For example, if you paid out $10 last month and took in $12, that's a profit of $2.  If it went the other direction, where you paid out $12 and took in only $10, that's a loss of $2, which is identical to a negative profit of $2.

Term of expression.  In the case of going from a loss of $100 to a loss of $25, that's a change in loss of 75%.  The old number is a loss, so you have to express the formula's result -- in this case a negative 75% -- in terms of a loss.  If the old term had been positive, i.e., a profit last month, then the result of the formula would have to be expressed in terms of a profit, not a loss, as you'll see below.  Note that, happily, this is true regardless of whether the new number is negative or positive.

It's always the old value that determines the term of expression,

not the new one.

So, we know the result of this formula must be expressed in terms of a loss, because the old term was a loss.

Increase or decrease?  Now, what do we do with the fact the result is a negative number?  The answer couldn't be simpler or more obvious:

If the result is negative, that's a decrease, and if it's positive that's an increase.

So, going from -100 to -25 means a decrease in loss of some percent.

When we apply the same, good old formula, it looks like this:

(New - old) divided by old

= (  [-25 ]  -  [-100 ]   )   divided by    (-100)

= (  75  )  divided by -100

= - 75%

So, since the answer is negative, the interpretation is that going from a loss of $100 to a loss of $25 is a decrease (because it's a negative number) in loss (because the old value was a loss) of 75%. 

Now we'll consider a increase from -100 to +25.   The formula results in an answer of minus 125%.  What this means is a decrease (because it's a negative number, remember) in loss, (it's expressed in terms of loss because that's the term of the old, or original, value).

So, going from -100 to +25 is a decrease in loss of 125%.

What about going from a loss of $25 to a loss of $100?  The formula still works.

Because the old term is a loss, we know that whatever the result is, it'll be expressed in terms of a change in the loss, not a change in the profit.

And because the answer, 300%, is a positive number, we know it's an increase, not a decrease.

In this case it's a 300% increase in loss.

Finally, the case of going from a profit of $25 to a loss of $100.  The formula results in -500%, which means, as if you didn't know by now, a 500% decrease in profit.


You may try out any of the examples above using the two little calculators below.  You enter your two numbers, and the software spells out the answer for you in complete detail.  Not only that, it's the correct answer.

Here some rules.

-- Change only the cells in the  yellow  background.  (You may mess with other cells such as by double-clicking, but if you're that interested you should go ahead and get the spreadsheet that incorporates both calculators and other useful information.  If you want the Excel spreadsheet behind these two calculators just .)

-- Ignore the bottom line of each calculator, the one in a black background with several icons.  It's basically advertising that pays the price for my ability to do this.  Or, you don't have to ignore it if you don't want to, but be prepared to be confused.

-- You may enter positive numbers, you may enter negative numbers, and you may enter zero, but since division by zero is nonsensical, if you see a #DIV/0! error message you should ignore that answer cell and read only the others.

-- You may enter any number of decimal places, but if there are too many, such as 123,456,789.012 the calculators get cranky and take it out on you by displaying the answers like this: #############.  This means the number won't fit horizontally in the space provided.  You can get back to normality by refreshing your browser, or you can try different numbers.

-- The precision of the display of numbers is the number of decimal places I chose, whether for good or ill, for particular cells.  If you want to be able to change those numbers and in every other way manipulate the whole spreadsheet, again you should  you want this.

The calculator immediately below takes as input the two numbers you want to compare to learn the percentage change.

The calculator now immediately below takes as input one of the two known numbers and the percentage and gives you the other number, again spelled out in complete detail.

Have fun.


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