Is a Powerball Ticket a Good Bet?

The advertised Powerball jackpot has now been revised up to $500 million and the game rules indicate that the chance of winning the jackpot is 1 in 175 million for each $2 ticket that is purchased.  To the naïve Powerball customer this may seem like the expected payout from a $2 ticket is well above $2.  That reasoning is incorrect because the $500 million payout over 30 years has a cash value of only $327 million and there is a 63% chance that a winning ticket will be shared with one or more other players.  If Powerball lottery officials are correct and the jackpot is $500 million on Wednesday night, the expected payout on a $2 ticket will be $1.53.  If lottery officials have underestimated Wednesday’s jackpot the expected value of a $2 ticket will be somewhat lower.  For example, if the jackpot is $550 million (paid out over 30 years) the expected value of a ticket declines to $1.50.

Lottery officials expect about 227 million $2 tickets to be sold.  32.5% of the proceeds from these ticket sales are added to the cash value of the jackpot with another 17.5% funding consolation prizes.  This suggests there should be a 50% return on a $2 ticket, or a $1 expected value based only on current sales.  The expected return on a $2 ticket is substantially higher than this, about $1.53, because of the players in the 15 previous drawings where no winning ticket was selected.  These prior Powerball players have subsidized players in this week’s drawing.

Although 227 million tickets are expected to be purchased between Saturday night’s drawing and Wednesday’s drawing, with the odds of a winning ticket being 1 in 175 million, there is a  37% chance that there will be no winning ticket on Wednesday night.  If even more than 227 million tickets are sold, the jackpot increases but so does the chance of sharing the prize with other winners.  The following table illustrates some possible outcomes if the jackpot continues to increase for the November 28th Powerball drawing.

Jackpot Values and Expected Returns for November 28 Powerball Drawing
Advertised Jackpot Cash Value of Jackpot Tickets Sold Chance of No Winning Tickets Expected Value of $2 Ticket
$500 million $327 million 176.3 million 36.6% $1.53
$550 million $360 million 226.7 million 27.4% $1.50
$600 million $393 million 277.0 million 20.6% $1.48

A $2 Powerball ticket typically has an expected return of $1 meaning that half of the money spent on tickets will not be paid out in prizes.  Because there have been 15 weeks of drawings with no winners the rollover in the Powerball jackpot is about $203 million.  This means a $2 ticket purchased this week has an expected value of $1.53.  It is more likely than not that ticket sales will be above the projected amount of 227 million, which will raise the jackpot but dilute the expected value of each ticket sold.  The expected value of a ticket will likely remain near $1.50, but as the jackpot rises it becomes much more likely that there will be at least one winner on Wednesday night.  A winner on Wednesday will reduce the advertised jackpot on Saturday December 1 to $40 million (cash value of $25.8 million).

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