Statistics, Probability, Lotteries and Dumb Programmers

Statistics and probability are basic mathematical concepts that are based on fact. The math itself may be challenging to some but it is quite exact and not subject to coercion, given a sufficiently large set with which to operate. This short article explains a pervasive myth about lotteries, and offers some calculations.

I remember distinctly some years ago, our developer group became involved in a discussion about probability regarding the toss of a coin. One developer, who shall remain unnamed, insisted that the prior results of a series of coin tosses would have a statistical influence on the result of a subsequent coin toss. Of course, this is a myth.

The result of any individual fair coin toss is completely unrelated to however many coin tosses preceded it. It is a completely isolated event. Yet, an entire industry of Lotto predictions and software depends on this myth. State lotteries publish historical results of previous games for just this purpose. Unwitting players buy software that will analyze these results and are supposed to be able to increase one's chances of winning by using their predictions and formulas. People (usually those that can least afford it) take money that should be going to rent or groceries and bet it on the Lotto in hopes of winning big.

The probabilities associated with a type 6/49 Lotto are determined by calculations based on equally likely outcomes but their interpretation is empirical. If an experiment (such as a weekly Lotto drawing) is repeated under similar conditions many times (mathematically the required number of times must approach infinity) then the probability of an event is the total proportion of experimental repetitions on which the event occurs.  For example, the probability of obtaining a fourth place prize is 0.0009686. This means that in 26,000 Lotto draws, each particular choice of six numbers will yield a fourth place prize (matching exactly 4 numbers) on approximately 0.0009686 of those plays; that is 26000 x 0.000986 = 26 times. There is no guarantee that this will happen 26 times. The actual number of times is random and subject to what is called the Poisson approximation. The figure of 26 represents an "expected" number of winnings and is a useful measure with which to compare the performance of different strategies.

How Probabilities are Calculated

The following discussion is a brief description of the process needed to derive Lotto 6/49 probabilities.

Jackpot (all six winning numbers selected)
There are a total of 13,983,816 different groups of six numbers which could be drawn from the set {1, 2, ... , 49}. To understand this, we observe that there are 49 possibilities for the first number drawn, following which there are 48 possibilities for the second number, 47 for the third, 46 for the fourth, 45 for the fifth, and 44 for the sixth. If we multiply the numbers 49 x 48 x 47 x 46 x 45 x 44 we get 10,068,347,520. However, each possible group of six numbers (combination) can be drawn in different ways depending on which number in the group was drawn first, which was drawn second, and so on. There are 6 choices for the first, 5 for the second, 4 for the third, 3 for the fourth, 2 for the fifth, and 1 for the sixth. Multiply these numbers out to arrive at 6 x 5 x 4 x 3 x 2 x 1 = 720. We then need to divide 10,068,347,520 by 720 to arrive at the figure 13,983,816 as the number of different groups of six numbers (different picks). Since all numbers are assumed to be equally likely (and state lotteries are fastidious about ensuring this) and since the probability of some number being drawn must be one, it follows that each pick of six numbers has a probability of 1/13,983,816 = 0.00000007151. This is roughly the same probability as obtaining 24 heads in succession when flipping a fair coin.

In other words, if I told you that you could invest 100,000 dollars with me and your expected return would be 7/10ths of one cent, would you consider the investment?  Yet, millions do this every day.

Second Prize (five winning numbers + bonus)

The pick of six must include 5 winning numbers plus the bonus. Since 5 of the six winning numbers must be picked, this means that one of the winning numbers must be excluded. There are six possibilities for the choice of excluded number and hence there are six ways for a pick of six to win the second place prize. The probability is thus 6/13,983,816 = 0.0000004291 which translates into odds against of 2,330,635:1.

Third Prize (five winning numbers selected, bonus number not selected)
As in the second prize there are six ways for a pick of six to include exactly five of the six drawn numbers. The remaining number must be one of the 42 numbers left over after the six winning numbers and the bonus number have been excluded. Thus there are a total of 6 x 42 = 252 ways for a pick of six to win the third prize. This becomes a probability of 252/13,983,816 = 0.00001802 or, equivalently, odds against of 55490.3:1.

Fourth Prize (four winning numbers selected)
There are 15 ways to include four of the six winning numbers and 903 ways to include two of the 43 non-winning numbers for a total of 15 x 903 = 13,545 ways for a pick of six to win the fourth prize, which works out to a probability of 13,545/13,983,816 = 0.0009686, that is odds against of 1031.4:1.

Fifth Prize (three winning numbers selected)
There are 20 ways to include three of the six winning numbers and 12,341 ways to include three of the 43 non-winning numbers for a total of 20 x 12,341 = 246,820 ways for a pick of six to win the fifth prize, which works out to a probability of 246,820/13,983,816 = 0.01765, that is odds against of 55.7:1.

You can easily calculate the odds against winning any lottery game with Excel. Just enter the total number of balls in cell A1, the number of balls you must pick in cell B1, and the formula =COMBIN(A1,B1) in cell C1. The number shown in that cell will be the odds against winning with a single ticket. If the payout for that particular game  is larger than this number, you have a statistical chance of a play being "worth the long run odds" - which of course, isn't much - but if you insist on playing at least you can determine whether or not to play a particular game based on the estimated payout.

The bottom line is that you are better off either burning your money, or putting it under your mattress. If you burn it, it will generate a bit of heat which can be useful in staying warm on a cold winter day. If you put it in a mattress, you can pull it out 20 years later and marvel at how much purchasing power it has lost. Buying many lottery draws at one time increases one's odds only very slightly, but basically represents throwing good money after bad.

Lotteries are a kind of voluntary tax. If you're like me, you probably already pay plenty in taxes and the last thing you need is to pay even more into a state lottery. If you want to improve state education, donate your money directly. A state lottery is a very inefficient way to help kids get a better education.

By Peter Bromberg   Popularity  (3618 Views)