EDIT: Looks like Shorena was faster at the math than me. Ah well, I'll leave my post here anyhow.
There are 292,201,338 unique ways to choose from the set of Powerball numbers. There is exactly 1 arrangement of those numbers that will win. Therefore the odds of winning the powerball with a single $2 set of numbers are a 1 in 292,201,338 of winning approximately $1 billion.
There are 1,461,501,637,330,902,918,203,684,832,716,283,019,655,932,542,976 unique bitcoin addresses. There can't possibly be more than 2,099,999,997,690,000 addresses holding a balance at any moment in time (if every potential bitcoin had already been generated, and they were distributed as 1 satoshi in each address).
If I've done the division correctly, that works out to the best possible odds anyone could theoretically ever have are about a 1 in 695,953,161,399,311,771,921,950,012,312,375 chance of "winning" 0.00000001 BTC
Odds would be multiplied for consecutive wins, so for someone who hadn't won the Powerball at all yet to be able to win twice consecutively with only a single $2 set of numbers chosen on each draw you'd be looking at a 1 in 85,381,621,928,990,244 chance of winning approximately $1.04 billion (the prize re-sets back to $40 million after the first win).
For 3 consecutive wins you'd be looking at a 1 in 24,948,624,168,261,090,285,746,472 chance of winning approximately $1.08 billion.
For 4 consecutive wins you'd be looking at a 1 in 7,290,021,363,225,027,714,833,921,447,179,536 chance of winning approximately $1.12 billion.
So, your odds of "winning" 0.00000001 BTC are better than your odds of winning a total of $1.12 billion through 4 consecutive Powerball draws, but not as good as your odds of winning a total of $1.08 billion through 3 consecutive Powerball draws.
Obviously bitcoin isn't spread quite as thinly as 1 satoshi per address, so realistically you're probably looking at better odds of an address collision than winning 5 consecutive Powerball drawings, but better odds of winning 4 consecutive Powerball drawings than encountering an address collision.
Note that this isn't just winning 2, 3, 4, or 5 Powerball draws in a lifetime (which is FAR more likely), this is consecutive draws.
Note also that the odds of "winning" powerball multiple times in a row doesn't change if you choose the same set of numbers every time. Therefore, another way of looking at those Powerball odds that I stated, is that it is the odds that the powerball draw will come up 1, 2, 3, 4, and 5 with a red ball of 6 that many times in consecutive draws.