Expected value tells you the return you can expect for some kind of action. For example, if you spend $10 on Powerball tickets every week, how much money can you expect to win or lose? We can answer that question with probability, which allows us to create a formula for Powerball expected value.
What we want to know here is the value of a ticket for an upcoming drawing. Expected value is a projected value based on the probability a set number of outcomes happening. “Projected” means that we can use data we know, like published odds for Powerball, to make a very good guesstimate about future values.
How to Calculate Powerball Expected Value
We need to know two pieces of information:
- The projected payout. This changes according to the number of players.
- The odds of winning: These odds are published by the Powerball lottery .
The math is as follows:
- Convert odds to probability:
- Place your chance of winning in the numerator (top) of a fraction.
- Add both values in the odds and place that value in the fraction’s denominator (bottom).
For example, odds of 1 to 32 becomes a probability of 1/33.
- Multiply probability by payout. For example, if you have a $4 payout and a 0.254 probability of a win, then 0.2543 * $4 = $0.10.
- Repeat for all payouts, excluding the jackpot.
- Add the individual expected values together.
I’m not factoring in the jackpot into these calculations because the odds of winning the jackpot are so tiny, that—statistically speaking—they are outliers. In other words, they are so very unlikely to happen that we can ignore them when calculating the Powerball expected value. In addition, the potential jackpots are highly variable (millions to billions), that makes any calculations we do practically worthless. If we did include a $1.56 billion jackpot in our calculations, the expected value of a $2 ticket would be $5.43, which is of course nonsensical. To add to the myriad of reasons why we shouldn’t add in the jackpot, it’s not awarded every week, making it a random factor “x“. You could place an x in your results for that random factor (i.e., expected value = $0.32x), but that would spoil the fun of the calculations, right?
Here are the results:
You can download the Powerball expected value worksheet here, which includes all of the calculations for with and without the jackpot.
With the cost of a ticket at $2.00 and an expected value of $0.32, you can expect to lose $2.00 – $0.32 = $1.68 on each ticket. In other words, if you play Powerball, you can expect to lose 84% of your money on every ticket that you buy, every time you play.
Other Factors to Consider
The above calculations are very basic, taking into account only stated odds and the cost of tickets. In real life, there are a myriad of other factors that will have an effect on Powerball expected value, including:
- Taxes: Depending on where you live, this could range from 25% to 35%.
- Whether you’ll claim a lump sum or one time payment.
- How you will invest your winnings: good investments may make the difference between a comfortable retirement or bankruptcy.
- Perhaps most importantly, it doesn’t factor in that you may have to share your winnings with other winners.
Stephanie Glen. "Powerball Expected Value" From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/powerball-expected-value/
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