powerball probabilities

[marbles]

With this Wednesday's jackpot at approx $160.0 mil, Powerball is once again approaching record levels. So it's a good time to brush up on our Powerball probabilities.

The rules of the game: five white balls are selected from a numbered set of 53, and one red ball (the Powerball) is selected from a numbered set of 42.

1. What is the probability of hitting the jackpot, matching all five white balls and the powerball?

2. If 60.0 million tickets are sold for this drawing, what is the probability that there is exactly one winning ticket? Two tickets? Three tickets? Zero tickets?

3. Here is the payoff schedule:
Match the Powerball (PB) only: $3
Match PB and one white ball: $4
Match PB + 2: $7
Match PB + 3: $100
Match PB + 4: $5,000
Match PB + 5: $160.0 Mil (important: jackpot is chopped in the event of multiple winning tickets, use numbers from #2, assume no more than 3 winners)
Match 3 white, no PB: $7
Match 4 white, no PB: $100
Match 5 white, no PB: $100,000.

What is the Expected payout on one ticket?