The math behind all the discussion of the Powerball drawing assumes true randomness – equal likelihood for each number to be chosen, both in the drawing itself and, crucially, in the process of assigning “Quick Picks” to ticket buyers who don’t wish to choose their own numbers.

Are those assumptions reasonable?

Imagine a bag filled with 10 red marbles and 20 blue marbles. Close your eyes, reach into the bag and pull out a marble. You might call your selection random, but more importantly, the choice of red or blue is not equally likely.

In the Powerball drawing, winning numbers are selected from two clear containers: one container has 69 white colored balls with each ball numbered in black ink with an integer from 1 to 69. The other container contains 26 red balls with each ball numbered in black ink with an integer from 1 to 26.

The balls are dropped into the respective containers and then mixed in the container by what appears to be air injected from the bottom of the container. The air is then turned off and a ball is raised from the bottom via a platform and then removed from the container. This procedure is repeated for the selection of each ball (five white and one red, the “Powerball”). Generally speaking, it seems reasonable that each ball is equally likely to be selected by this process.

It is possible – though it’s a stretch – that balls with printed numbers requiring more ink to delineate the number on the ball may weigh more due to the extra ink than balls requiring less ink. Coupled with gravity, this may be enough to keep those balls lower in the container and thus more likely to be picked by the platform. In short, the ball marked 68 may be more likely to be picked than the ball marked 1.

Luckily, this is a testable assumption. Studying the results from previous drawings would allow an assessment of whether each number is occurring with similar frequency. Without doing the statistical calculation and data collection, given the nature of this device for generating balls/numbers, it’s safe to assume that this process generates each number with equal probability.

Evaluating the “Quick Pick” numbers is more challenging. Without a machine to generate numbers with plastic balls, lottery machines nationwide have been generating numbers for ticket buyers in ways that may not give each number exactly equal chances of being chosen.

The potential problems come from the fact that computers are devices programmed by humans and so, almost paradoxically, they must be given a systematic method to choose random numbers. In computerprogramming terminology, this is often called generating a “pseudo random” number.

In this process, the computer may use some information, such as the computer’s real time clock with precision to a millisecond, at the time that a request for a lottery ticket was made, to trigger a process that draws five numbers and one powerball number. This beginning number is often called the “seed.” Other seeds may be created from different phenomena that presumably occur without reason or predictability. From those seeds, additional calculations generate numbers at rates that approximate randomness.

The randomness of these machines’ results can also be tested, but with more difficulty: it involves either buying large numbers of “Quick Pick” tickets or collecting ticket information from a large number of people. Analyzing the frequencies of the numbers that were generated would reveal the degree of randomness of the Quick Pick process.

Without these data, it can be illuminating to look at the number of Powerball tickets sold and the percentage of the 292,201,338 possible combinations that are covered by those tickets. These data strongly suggest that the Powerball computers are generating combinations with equal probability and thus at random.

In conclusion, it appears we have both mechanisms operating randomly and are free to compute the odds of winning, probability that there’s at least one winner, and, most importantly, our expected profits.

source: livescience.com by Jeffrey Miecznikowski