When & How Should you Play the PowerBall Lottery?

Do you play the PowerBall lottery on a consistent basis?  Why not improve your odds of making money by following some simple principles about when and how you should play PowerBall.

If you're unfamiliar with PowerBall, here's a quick run-down on how to play and your odds of winning.  Every Wednesday and Saturday, 5 white balls are selected at random from a group of 59, and a single red PowerBall is selected from a group of 35.  To win the jackpot (which is always a minimum of $40,000,000), you have to match the 5 white balls and the 1 red PowerBall.  However, in addition to the jackpot, there are eight other ways to win, each with a different cash winning and chance (odds) of winning; here are the ways to win:

powerball.com

To play, each ticket costs you $2.  There is a 'multiplier' option with your ticket that costs an additional $1, and it works like this: all winnings (except for the jackpot) are multiplied by a number (2 through 5) chosen by a computer prior to the drawing of balls (note that the higher numbers are selected less often by the computer).  The winning with the multiplier are as follows:

powerball.com

Great, you know how to play - but should you?
To help us answer this question, we first need to think of your ticket purchase as an investment, and the drawing outcome as either your profit or loss.  In the simplest case, you make an investment of $2 and hope for a profit.  If you win, you profit is the winnings minus the ticket cost (e.g. $1,000,000 - $2 = $999,998).

We know the odds of each type of win, and the respective payout for each (we'll assume NO multiplier for now).  With this information, we can calculate your ticket's expected value.  Expected value is the average profit (or loss) you expect to make if you play on a continual basis.  If you play only once, your ticket will either be worth $0 (no win) or greater than $0 (you won something), obviously NOT the average.  However, if you consistently play, you will win some times and lose other times, and you'll have an average profit (or loss) depending on your tickets' performances.  This average profit/loss is expected value.  If we assume a jackpot size of $40,000,000 (the minimum), you have the following expected value for your ticket:

The above table tells us that by consistently buying $2 tickets, you can expect a loss in the long run, if the jackpot is $40,000,000.  Put another way, for each ticket you buy, your investment of $2 will, on average, reap a $1.41 loss.  Not good.  You might as well put that money towards paying your NYC rent.

But we also know that the jackpot size increases over time if there are no winners.  So there must be some jackpot size that is large enough to make the investment break-even.  This is true, and the below graph shows the jackpot size needed to achieve this:

FIGURE #1

The above graph tells us that you shouldn't play PowerBall unless the advertised jackpot size is greater than $290,000,000.  Why?  Because this jackpot is large enough to compensate for the dismal odds of winning, and you'll thus have a positive investment in the long run.

But there's a big catch
What if there are multiple jackpot winners?  If multiple people win a jackpot, the winnings are split across all recipients.  This outcome would significantly reduce your ticket's value.  We have to include the probability of more than one jackpot winner within the expected value calculations made above.  To do this, we leverage what are called Bernoulli trials.  In a nutshell, Bernoulli trials calculate the probability of N number of successes with X number of attempts in a particular scenario.  The scenario is winning the jackpot, the number of success is 'greater than 1 winner' and the number of attempts (i.e. trials) is the number of purchased tickets.  The following graph, created using the equation for Bernoulli trials, shows the probability of 0 winners, 1 winner and 'more than 1 winner' as the number of participants increases:

If the graph is complicated, don't worry about it.  The take away is that the probability of more than one winner increases as more lottery tickets are sold (intuitive, right?) but this, unfortunately, decreases your ticket's expected value.  How much does this change your ticket's expected value?  In other words, how will this change Figure #1 above?  To answer this question, we need to know how many people will participate in the lottery at different jackpot sizes.  Luckily for us, there is a strong correlation between the jackpot size and the number of tickets sold.  The below graph (built from two years of historical PowerBall data) clearly shows this correlation: as the jackpot increases in size, more people are struck with lottery fever, and more tickets are sold.  Note I have also placed a regression model line on top of the data, and it will be used to estimate the number of players based upon the jackpot size.

lottoreport.com

Now we have everything needed to update Figure #1 and thus better understand your ticket's expected value with the probability of multiple jackpot winners.  The below chart is the updated version of Figure #1, and it incorporates everything we've learned thus far (odds of you winning, ticket expected value, and probability of other players winning).  As you can see, the below chart has an interesting shape: your ticket's expected value is positive for a certain range and then becomes negative again.  Specifically, your ticket's expected value is positive if the jackpot size is between the values of $300,000,000 and $570,000,000:

FIGURE #2

This means that if you play on a consistent basis ONLY when the jackpot is between $300 and $570MM, you will have, on average, a positive return.  Put another way, for every $2 you spend to purchase a PowerBall ticket, you can expect up to a $0.50/ticket return on your investment.

Cash-Out & Multiplier Options
To be clear, everything we've covered thus far assumes a regular $2 ticket (with no multiplier option at $1 extra per ticket) and the Annuity Option if you win the jackpot.  The annuity option means you win the advertised jackpot amount and are paid it over the course of 30 years (one payment a year).  However, most people choose the 'cash-out' option, meaning the jackpot winnings are immediately paid-out; the consequence is that the cash-out amount is significantly reduced in comparison to the advertised / annuity amount.  I looked at all the winning amounts for the past several years and, on average, the amount is reduced by factor of 1.85.  For example, your $40,000,000 annuity winning would be reduced to approximately $21,620,000 if you opted for the cash-out option.

Now figure #2 above can be updated with the expected value of your ticket under two new scenarios: (1) if you choose the cash-out option, and (2) if you choose the multiplier options (note, I have assumed the best multiplier option of 5).  The results are as follows in Figure #3 below.  As you can see, in the long run, the cash-out option as well as the multiplier option (even in the best case scenario) do NOT result in positive investments:

FIGURE #3

Recommendations
Your chances of winning are the same for every ticket you purchase (and the odds are dismally small, let's just be honest with ourselves) but if you consistently buy PowerBall tickets, you might as well improve your return on investment buy following these recommendations:

• Purchase tickets only when the advertised jackpot is between $300 and $570MM
• Purchase only the regular $2 ticket without the multiplier option
• Choose the annuity option for collecting your jackpot winnings

Best of luck!

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Excel was used for this analysis; spreadsheets available upon request
Source of data and pictures from Powerball:

http://www.powerball.com/powerball/pb_stories.asp

http://www.powerball.com/powerball/pb_prizes.asp