The math behind Powerball X’s & O’s

Powerball X’s & O’s is a simple lottery game offering a rolling jackpot and fixed prizes. It uses the logos of all 32 NFL teams instead of numbers.

Category

Powerball X’s & O’s is a combinatorial lottery game drawing from one pool: eight logos drawn from 32.

Size of the draw: 8.

Size of a played line: 8.

Cost of a line: $5

Its parametric design is equivalent to an 8/32 number lottery matrix (which currently does not exist), and the associated odds and statistical indicators are the same regardless of the nature of the objects being drawn (numbers, logos, or something else).

Hence, the game is unique within its particular category, in terms of either the size of the draw or of the pool.

Prizes

There are five tiers of prizes in the draw, defined as matching 4 to 8 logos. They are all fixed prizes, except the top prize (8 logos matched), which is a rolling-over jackpot.

The prizes do not increase uniformly from the lowest to the highest. There is an abrupt jump between the first two tiers, i.e., from jackpot to $20,000, and from the second to third tier, i.e., from $20,000 to $500, also for the corresponding odds of winning.

The odds of winning

The odds of winning with one line range from 1 in 10,518,300 (for the jackpot) to 1 in

14.14 (for the lowest tier). The overall odds of winning any prize are 1 in 12.15.

Average win and fairness

The next table lists the average win for each prize:

For a jackpot of $1 million, the average win is $0.095072.

Example of use and interpretation of the table: Over the long run, a win in the 5 of 8 prize tier is distributed as $0.431035 average win per line played (assuming the prize is won with the relative frequency of about 1 in 93 draws).

The next table lists the fairness of each prize relative to its probability:

For a jackpot of $1 million, its fairness would be 1.90%. However, this only applies in the circumstances where such a jackpot is actually won, as rollovers actually mean zero prizes in this tier.

These values indicate a far higher fairness of these prizes than in the other lotteries that we analyzed.

Expected value and house edge

The precise expected value of the Powerball X’s & O’s game cannot be obtained, as the first-tier prize is variable. We will estimate it in two cases:

a) The first case is taking the jackpot amount as zero (case justified by a long series of draws when the jackpot is not won, given its probability).

b) The second case is taking the jackpot amount as $1 million (just as a reference, based on the operator’s estimation for the first draw when the game will be launched).

In case a), the expected value is –$3.34, or –66.83% as a percentage of the line cost. The house edge of the game is 66.83%.

In case b), the expected value is –$3.24, or –64.93% as a percentage of the line cost. The house edge of the game is 64.93%.

As an interpretation, in case a) you are expected to lose on average $3.34 at every $5 invested, over the long run.

Final remarks

Powerball X’s & O’s offers higher odds of winning than the lotteries in the usual combinatorial categories (5-6 as the size of the draw and 40-60 numbers in the pool). Yet this does not make it suitable for syndicating, as the higher cost of a line and a low second-tier prize relative to its probability counterbalance the odds advantage.

It is remarkable that the prizes in this game have a far higher fairness than in most of the lotteries; however, its house edge does not jump far lower than in the other lotteries.

These facts make the game a good option for chasers of the jackpot and the second-tier prize in isolated plays rather than for players looking for small profits over the long run.