Today is the big Powerball drawing, with an estimated jackpot of 1.5 billion dollars. This has triggered the usually lottery mania, as people who do not normally play plunk down their money as though the larger jackpot makes the risk more acceptable. Tomorrow they’ll tune in, with visions of easy riches dancing through their head. And, likely, one or more will win the jackpot.
The problem is that most won’t . The odds of winning are small, indeed. Or, to look at it another way, the odds of losing are practically a sure thing. But the odds are simple probability theory, and like other branches of mathematics, it functions regardless of wishful thinking. Professional gamblers know this, just as they know the odds on their particular games of chance, and make their decisions based on probability.
It should come as no surprise that probability theory has its roots in gambling. Gamblers have long attempted to analyze games of chance, looking for possible advantages. In the 16th Century, the Duke of Tuscany approached Galileo with a question on the probability of dice, and Galileo answered his question by making a table of the 216 possible throws with three dice. But probability as a branch of mathematics had to await the 17th Century, when French nobleman Antoine Gombaud asked Blaise Pascal a question about dice, and that prompted Pascal to write fellow mathematician Pierre de Fermat, and their exchange of letters laid the foundation of basic probability theory. For while individual games had long been analyzed, there was no general theory of probability.
When their work came to light, this sparked the interest of gamblers everywhere, and such was the interest that this new branch of mathematics rapidly developed. Long before the 21st Century, mathematicians – and gamblers – understood probability well enough that casinos could set up honest games where the odds slightly favored the house. This, of course, meant that casinos could turn a profit.
Lotteries are no different. Whether it’s state-run outfits or illegal numbers runners, the odds favor the house, with prizes (usually) less than the value of tickets purchased. In the hands of the states, it essentially forms an optional tax. Given the odds, Voltaire called lotteries a tax on stupidity.
Depending on the lottery, the odds can be long, indeed. Powerball has players select five numbers from a pool of 69, with the sixth Powerball number drawn from a separate pool of 26. Fortunately, you don’t have to chose the numbers in order, but what kind of odds are we talking about?
To calculate the odds, let’s first look at the odds of picking one number in 69 correctly. With only one draw, you have just one chance of guessing, or 1 in 69. We could note this as (69/1).
Now let’s look at two numbers. That means two draws out of a pool of 69 numbers, or (69/2). So, the odds of getting two numbers out of 69 correct is 69/2 x 68/1. Notice that both the pool of numbers, at the top, and the number of the draw, at the bottom, are decreased by 1. When we multiply this out, we have (60 x 8)/(2 x 1) = 4,692/2 = 2,346/1. This means the odds are 1 in 2,346 of getting two numbers right.
We use the same method to calculate the odds for the remaining numbers. For three correct numbers, the odds are 1 to 52,395; for four, 1 to 864,501; and for five 1 to 11,238,513. That’s 69/5 x 68/4 x 67/3 x 66/3 x 65/1 = 1,348,621,560/120 = 11,238,513/1, or 1 to 11,238,513.
This leaves us with the Powerball number. Since the Powerball is pulled from a separate group of 26 numbers, that means there’s a 26/1 chance of guessing the correct number. Notice that we calculated the odds for the first five by multiplying the odds of each draw, so, to calculate the Powerball jackpot odds, we multiply both odds together: 11,238,513/1 x 26/1 = 292,201,338/1, or 1 to 292,201,338 of winning.
If you check the actual Powerball odds, you’ll find it slightly different for the white balls, and different for picking the Powerball alone. That’s because game play requires picking only the Powerball, or, only a certain number of the white balls with or without the Powerball, and requirements such as none of the white balls equal the Powerball changes the odds slightly. But for jackpot play, the odds agree with those from Powerball.
Knowing the odds doesn’t necessarily mean understanding what they mean. For instance, one scheme for picking lottery numbers is based on the premise that he odds of any one ball coming up is the same, and so if one doesn’t seem to come up as often, then it’s “overdue.” The fallacy here is that each drawing doesn’t affect the other. Each Powerball drawing has exactly the same odds.
Another scheme is picking hot number, numbers that come up more frequently, or even using self-learning programs called neural networks to predict likely numbers. Unlike the idea of overdue numbers, this at least does have a better premise. Each ball is made to exacting tolerances, and is weight to check for variations, but there is still going to be minute differences, and balls become worn during play, and all of this affects which ball is likely to come up. It’s slight, but still a factor. So, at least in theory, tracking hot numbers or looking for patterns with a neural network might give an edge.
The problem is that those who run the lotteries know this. For this reason, lotteries use more than one set of balls, each set chosen at random. In addition, balls are replaced regularly due to wear. This defeats any sort of statistical analysis or neural network by introducing additional random factors even before the set is replaced. And once a different set is use, or is replaced, any sort of frequency analysis or neural network is going to go right out the window.
All this means that when the Powerball drawing is held, what numbers come up is going to be pretty close to purely random, close enough that anyone who does come up with the numbers, either through guessing or using a pseudorandom generator, or whatever scheme they chose, will do so purely by chance. That’s a 1 in 292,201,338 chance, regardless of the size of the jackpot.
At those odds, I think I’ll pass.