Abrazolica

The powerball jackpot is now over a billion dollars. The idea of winning that kind of money is too crazy for anyone to ignore, so let's do a little analysis of the game.

To play the game, you fill out a ticket with 5 numbers ranging from 1 to 69, no duplicates allowed, and one number from 1 to 26. If you're lazy you can let the computer randomly pick the numbers for you. In any case, each ticket will cost you 2 dollars and give you one chance to win one of nine prizes that range from 4 dollars up to the grand prize.

How many possible tickets are there? You can choose 5 numbers in \(69\cdot 68\cdot 67\cdot 66\cdot 65 = 69!/64! = 1,348,621,560\) different ways but the order of the numbers doesn't matter so this has to be divided by \(5!=120\), giving

\[\frac{69!}{5! 64!} = \binom{69}{5} = 11,238,513\]

for the number of ways to choose 5 different numbers from 1 to 69 where the order doesn't matter. Note that I used binomial coefficient notation

\[\binom{n}{k}=\frac{n!}{k! (n-k)!}\]

which counts the number of ways \(k\) objects can be chosen from \(n\) when the order doesn't matter.

To complete the ticket you still have to choose one number from 1 to 26, which can obviously be done in 26 different ways. The total number of possible tickets is then

\[26\binom{69}{5} = 292,201,338\]

That's over 292 million tickets and only one of them will get you the grand prize. These are very long odds but hey, it's only 2 dollars for a nonzero chance at becoming a billionaire. Who can resist?

What about the other prizes? The table below shows all the prize amounts in dollars and the number of matches needed to win. The 5 numbers are randomly chosen by drawing balls from a drum containing 69 white balls. The single number, called the powerball, is randomly chosen from a drum containing 26 red balls. In the table, the \(wb\) column is the number of white ball matches and the \(pb\) column indicates the powerball match.

Matching \(k\) white balls means matching \(k\) of the 5 drawn balls and matching \(5-k\) of the other 64 balls. The number of ways this can happen is

\[\binom{5}{k}\binom{64}{5-k}\]

To get the probability, divide this by the number of ways 5 white balls can be drawn from 69.

\[P(wb=k)=\frac{\binom{5}{k}\binom{64}{5-k}}{\binom{69}{5}}\]

If you're familiar with discrete probability distributions then you may recognize this as a Hypergeometric distribution. In general, a hypergeometric distribution applies when there is a population of size \(N\) that contains a sub-population of size \(K\). The elements of the sub-population are called successes and we want to know the probability of getting \(k\) successes when sampling \(n\) elements from the general population. That probability is given by

\[P(k)=\frac{\binom{K}{k}\binom{N-K}{5-k}}{\binom{N}{n}}\]

For powerball, set \(N=69\), \(K=5\), and \(n=5\) to get the above equation for \(P(wb=k)\). The mean of a Hypergeometric distribution is given by

\[\overline{k}=n\frac{K}{N}\]

which for powerball is \(\overline{k}=\frac{25}{69}= 0.3623\). If you play a lot of powerball, this is the average number of white balls you will match.

We still need the probability of matching the powerball. There is only one way to match and there are 26 powerballs so the probability of matching is \(p=1/26\). The probability of not matching is \(1-p=25/26\), therefor

\[P(pb=m)=p^m(1-p)^{(1-m)}\]

The prize probabilities can then be calculated from the formula

\[P(wb=k,pb=m)=\frac{\binom{5}{k}\binom{64}{5-k}}{\binom{69}{5}}p^m(1-p)^{(1-m)}\]

The probabilities are shown in the following table.

All these probabilities are very low but there is a way to play that will guarantee you win at least something. If you plan to buy at least 26 tickets then be sure to play all 26 of the powerball numbers. This will guarantee a win of at least 4 dollars so you essentially get 26 tickets for the price of 24.

The most likely result is that you match no white balls and no powerball. The probability of this happening is \(0.6523337685743246\). If you play a lot of powerball then over \(65\) percent of the time you will match nothing. Note also that matching 2 white balls and no powerball or 1 white ball and no powerball gets you nothing. The probabilities for these matches are \(0.03564665402045489\) and \(0.2718057369059686\) respectively.

If powerball had a constant grand prize, represented by the variable \(gp\), and you played it repeatedly, how much could you expect to win on average per game? This is called the expected return. To calculate it, subtract the 2 dollar ticket price from each of the prize values, multiply by the respective probabilities and take the sum. The result is

\[\frac{gp-490936628}{292201338}\]

For a grand prize less than \(490,936,628\) the expected return is negative. If the grand prize is larger than \(491\) million dollars then the expected return becomes positive. Of course the grand prize fluctuates in value and you can't play often enough for this expectation calculation to make any sense because most of the probabilities are so small. Still I use it as a rule of thumb for when to play. If the grand prize gets above \(491\) million, I buy a ticket, just one, otherwise I leave it alone.