How is this expected value calculated?
Expected value (EV) is the sum of the probabilities of outcomes occurring, multiplied by the results of those outcomes.
A simple example
Suppose you roll a four-sided die:
- If it lands on 1, you get $1
- If it lands on 2 or 3, you get $2
- If it lands on 4, you get $0
Each face has a probability of 0.25.
The expected value of one roll is:
0.25(1) + 0.50(2) + 0.25(0) = 1.25
So the EV of this game is $1.25.
Playing this game for $1 would be mathematically worth it, while playing it for $2 would be mathematically not worth it.
EV does not predict what will happen on a single play, it just calculates what the payout will be over a large number of plays.
Applying this to Powerball
Powerball works the same way, just with many more possible outcomes.
The expected value of a Powerball ticket has two components:
- Lower-tier prizes
- The jackpot
Lower-tier prizes
If you add up the probabilities of all non-jackpot prizes and multiply each by its payout, the total expected value of the lower-tier prizes is:
$0.32 per ticket
This value:
- Does not change with the number of players
- Does not depend on the jackpot size
- Is fixed by the game rules
The jackpot
The jackpot is more complicated because it may be split between multiple winners.
To calculate the jackpot EV, we consider:
- The probability that you win the jackpot
- The probability that zero, one, two, or more other players also win
- The share of the jackpot you receive in each case
If you win:
- As the sole winner, you receive the full cash value
- With one other winner, you receive half
- With two other winners, you receive one-third
- And so on
The number of other jackpot winners is modeled using a standard Poisson approximation, based on the number of tickets sold.
Cash value and taxes
All jackpot calculations use the cash value only. The annuity and the time value of money are not considered.
Federal taxes are applied as a single effective rate. State tax is not included by default, but can be adjusted in the calculator.