Hack 41. Play Smart When You Play the Lottery

Powerball Payoff

Armed with all the wisdom you likely now have as a statistician (unless this is the first hack you turned to in this book), you might have already made a few interesting observations about this payoff schedule.

Easiest prize

The easiest prize to win is the powerball only match, and even then there are slim chances of winning. If you match the powerball (and no other numbers), you win $3. The chances of winning this prize are about 1 in 69.

This is not a good bet by any reasonable standard. It costs a dollar to buy a ticket, to play one time, and the expected payout schedule is $3 for every 69 tickets you buy. So, on average, after 69 plays you will have won $3 and spent $69.

Actually, your payoff will be a little better than that. The odds shown in Table 4-14 are for making a specific match and not doing any better than that. Some proportion of the time when you match the powerball, you will also match a white ball and your payoff will be $4, not $3. Choosing five white ball numbers and matching at least 1 will happen 39 percent of the time.

So, after having matched the powerball, you have a little better than a third chance of hitting at least one white ball as well. Even so, your expected payoff is about $3.39 for every $69 you throw down that rat hole (I mean, spend on the lottery), which is still not a good bet.

Powerball only

The odds for the powerball only match don't seem quite right. I said there were 42 different numbers to choose from for the powerball, so shouldn't there be 1 out of 42 chances to match it, not 1 in 69?

Yes, but remember this shows the chances of hitting that prize only and not doing better (by matching some other balls). Your odds of winning something, anything, if you combine all the winning permutations together are 1 in 37, about 3 percent. Still not a good bet.

Grand prize

The odds for the grand prize don't seem quite right either. (Okay, okay, I don't really expect you to have "noticed" that. I didn't either until I did a few calculations.)

If there are 5 draws from the numbers of 1 to 55 (the white balls) and 1 draw from the numbers 1 to 42 (the red ball), then a quick calculation would estimate the number of possibilities as:

In other words, the odds are 1 out of 21,137,943,750. Or, if you were thinking a little more clearly, realizing that the number of balls gets smaller as they are drawn, you might speedily calculate the number of possible outcomes as:

But the odds as shown are somewhat better than 1 out of 1.7 billion. The first time I calculated the odds, I didn't keep in mind that the order doesn't matter, so any of the remaining chosen numbers could come up at any time. Hence, here's the correct series of calculations:

Winning Powerball

OK, Mr. Big Shot Stats Guy (you are probably thinking), you're going to tell us that we should never play the lottery because, statistically, the odds will never be in our favor. Actually, using the criteria of a fair payout, there is one time to play and to buy as many tickets as you can afford.

In the case of Powerball, you should play anytime the grand prize increases to past $146,107,962 (or double that amount if you want the lump sum payout). As soon as it hits $146,107, 963, buy, buy, buy! Because the chances of matching five white balls and the one red ball are exactly one out of that big number, from a statistical perspective, it is a good bet anytime your payout is bigger than that big number.

For Powerball and its number of balls and their range of values, 146,107,962 is the magic number. The idea that your chances of winning haven't changed but the payoff amount has increased to a level where playing is worthwhile is similar to the concept of pot odds in poker [Hack #37].

You can calculate the "magic number " for any lottery. Once the payoff in that lottery gets above your magic number, you can justify a ticket purchase. Use the "correct series" of calculations in our example for Powerball as your mathematical guide. Ask yourself how many numbers you must match and what the range of possible numbers is. Remember to lower the number you divide by one each time you "draw" out another ball or number, unless numbers can repeat. If numbers can repeat, then the denominator stays the same in your series of multiplications.

One important hint about deciding when to buy lottery tickets has to do with determining the actual magic number, the prize amount, which triggers your buying spree. The amount that is advertised as the jackpot is not, in fact, the jackpot. The advertised "jackpot" is the amount that the winner would get over a period of years in a regular series of smaller portions of that amount. The real jackpotthe amount you should identify as the payout in the gambling and statistical senseis the amount that you would get if you chose the one lump sum option. The one lump sum is typically a little less than half of the advertised jackpot amount.

So, if you have determined that your lottery has grown a jackpot amount that says it is now statistically a good time to play, how many tickets should you buy? Why not buy one of each? Why not spend $146,107,962 and buy every possible combination? You are guaranteed to win. If the jackpot is greater than that amount, then you'll make money, guaranteed, right? Well, actually not. Otherwise, I'd be rich and I would never share this hack with you. Why wouldn't you be guaranteed to win? The probably is that you might be forced to...wait for it...split the prize! Argh! See the next section...

Don't Split the Prize

If you do win the lottery, you'd like to be the only winner, so in addition to deciding when to play, there are a variety of strategies that increase the likelihood that you'll be the only one who picked your winning number.

First off, I'm working under the assumption that the winning number is randomly chosen. I tend not to be a conspiracy theorist, nor do I believe that God has the time or inclination to affect the drawing of winning lottery numbers, so I'm going to not list any strategy that would work only if there were not randomness in the drawing of lottery numbers. Here are some more reasonable tips to consider when picking your lottery numbers:

Let the computer pick

Let the computer do the picking, or, at least, choose random numbers yourself. Random numbers are less likely to have meaning for any other player, so they are less likely to have chosen them on their own tickets. The Powerball people report that 70 percent of all winning tickets are chosen randomly by the in-store computer. (They also point out, in a bit of "We told you that results are random" whimsy that 70 percent of all tickets purchased had numbers generated by the computer.)

Don't pick dates

Do not pick numbers that could be dates. If possible, avoid numbers lower than 32. Many players always play important dates, such as birthdays and anniversaries, prison release dates, and so on. If your winning number could be someone's lucky date, that increases the chance that you will have to split your winnings.

Stay away from well-known numbers

Do not pick numbers that are well known. In the big October 2005 Powerball results, hundreds of players chose numbers that matched the lottery ticket numbers that play a large role in the popular fictional TV show Lost . None of these folks won the big prize, but if they had, they would have had to divide the millions into hundreds of slices.

There is also a family of purely philosophical tips that have to do with abstract theories of cause and effect and the nature of reality. For example, some philosophers would say to pick last week's winning numbers. Because, while you might not know for sure what is real and what can and cannot happen in this world, you do know that, at least, it is possible for last week's numbers to be this week's winning numbers. It happened before; it can happen again.

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Though your odds of winning a giant lottery prize are slim, you can follow some statistical principles and do a few things to actually control your own destiny. (The word for destiny in Italian, by the way, is lotto. ) Oh, and one more thing: buy your ticket on the day of the drawing. If too much time passes between your purchase and the announcement of the winning numbers, you have a greater likelihood of being hit by lightning, drowning in the bathtub, or being struck by a minivan than you do of winning the jackpot. Timing is everything, and I'd hate for you to miss out.