Odds of Winning the Lottery Using the Same Numbers Repeatedly Better/Worse? 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.
 A: No. Since the lottery numbers are randomly generated each time independently of the previous draws, the probability of winning is always the same.
A: 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.)
A: The probability of winning does not depend on the specific numbers selected and the numbers drawn for each lottery drawing have no dependence on previous drawings, so there is no benefit to playing the same numbers every time.
For most U.S. state lotteries, it is beneficial to choose numbers above 31 when possible as many people play numbers based on dates and so picking numbers above 31 lowers the likelihood of split pots.  But, really, playing the lottery is a losing proposition regardless of the mechanics.
A: No, unless you know the exact probability that each number will come out; in that case, you should always choose the most probable numbers.
Another case in which you should choose the same numbers if you have some extra information is in Isaac's answer, but that only applies if you change your goal to maximize your gain (or rather, minimize your loss) rather than maximize the probability of winning.
A: I have nothing to add to Qiaochu's and Isaac's answers and comments on the probability of winning. If you are buying tickets to maximize the earnings if you happen to win then the selection of numbers becomes important. It is best to try to pick numbers that other people are unlikely to pick. Examine the numbers when no one wins or only one person wins and try to find a property that many of these numbers have in common. If you find such a property then examine numbers when there were multiple winners and see if few of these numbers satisfy the property. Some years ago, when the rules were somewhat different than today I noticed that there were very few winners when the numbers contained consecutive integers. I do not know if this is still a good method for choosing numbers that fewer other people choose.
