Does the probability of winning the lottery differ between randomly generated numbers vs. selecting the same numbers every time?
Specifically. I'm interested in a breakdown of the odds per number for a given set of numbers that comprise a single US Powerball drawing (five white numbers plus the one powerball number), and how they arrive at the odds seen here: http://www.powerball.com/powerball/pb_prizes.asp
Tying that back to my original question, I was interested if playing the same numbers every drawing changes those odds.
Best Answer
So here is how Powerball works. You choose five different numbers between $1$ and $59$ inclusive (the white balls) and one number between $1$ and $39$ inclusive (the red ball). If the white balls match the winning numbers for the white balls, in any order, and if the red ball matches the winning number for the red ball, then you win the jackpot.
Because you can match the white balls in any order, the Powerball winning numbers are usually presented from smallest to largest, so if you order your numbers from smallest to largest, the two sequences have to match. The number of ways to pick five different numbers in any order from $1$ to $59$ is
$$\frac{59 \cdot 58 \cdot 57 \cdot 56 \cdot 55}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = 5006386$$
and every choice of five different numbers in increasing order has the same probability (one over the above number) of being chosen. One way to get the above number is as follows: first, pretend that order matters. Then there are $59$ possibilities for the first number. Since there are $58$ possibilities left, there are $58$ possibilities for the second number. And so forth. But since order doesn't matter, you can draw any set of five numbers in $5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ different ways (five factorial), so you have to divide by that.
Matching the red ball is easy: there are $39$ choices, so you have a $\frac{1}{39}$ chance of doing it. So, in summary, your odds of winning the jackpot from any choice of numbers is
$$\frac{1}{5006386 \cdot 39} = \frac{1}{195249054}$$
just as reported on the Powerball website. In other words, it's one over the total number of possible tickets. (The other probabilities reported on the website are slightly harder to calculate, but not by much; for that you need to learn about something called the inclusion-exclusion principle.)