The Real Jackpot Payoffs for Powerball on November 28th

Powerball is advertising that the expected jackpot for tonight’s drawing is $500 million and that there is a 1 in 175 million chance of matching all six numbers and winning a share of the jackpot.  The ads are misleading because $500 million equals the undiscounted value of payouts received over a 30 year period, not the cash value, and assumes that the prize will not be shared.  A more accurate description of the jackpot prize distribution is:

• A one in 476 million chance of winning $163.7 million (the jackpot is shared by two winners)
• A one in 479 million chance of winning $327.4 million (a single jackpot winner)
• A one in 947 million chance of winning $109.1 million (the jackpot is shared by three winners)
• A one in 2.82 trillion chance of winning  $81.9 million (the jackpot is shared by four winners)
• A one in 11.23 trillion chance of sharing the jackpot among at least five winners

If the advertised jackpot increases to $600 million, a more accurate description of the jackpot prize distribution would be:

• A one in 539 million chance of winning $196.4 million (the jackpot is shared by two winners)
• A one in 682 million chance of winning $131.0 million (the jackpot is shared by three winners)
• A one in 852 million chance of winning $392.9 million (a single jackpot winner)
• A one in 1.29 trillion chance of winning  $98.2 million (the jackpot is shared by four winners)
• A one in 3.27 trillion chance of sharing the jackpot among at least five winners

The most likely outcomes are that there will be a single winning ticket or the jackpot will be shared by two winners.  As the advertised jackpot nears $600 million, it becomes much more likely that there will be at least a three-way split of the jackpot.  Either way, the best strategy for lottery players is to choose a combination of six numbers that the other 200+ million lottery players won’t select.

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 28^th 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).

Why a Mega Millions Ticket is a Good Bet, But… (Revised 03.30.2012)

The Mega Millions multi-state lottery is now advertising a jackpot of $640 million for tonight’s drawing. The cash value of the jackpot is $460 million, a more accurate estimate of the present value of the prize. The jackpot has accumulated because there has been no winner for 18 consecutive drawings – since January 24th. This means that those of us who buy Mega Millions tickets for the first time this week have been subsidized by the millions of players who contributed to the jackpot over the past two months.

The probability of correctly selecting all six numbers for Friday’s drawing is one in 175 million. It is likely that there will be multiple jackpot winners. Even after taking this into account, the expected value of a one dollar Mega Millions ticket is about 90 cents.

About 18 cents of every dollar collected from Mega Millions sales are used to pay consolation prizes. Another 32 cents of each dollar goes into the cash jackpot. The cash value of Friday’s drawing is estimated to be about $200 million higher than in Tuesday’s drawing. In other words lottery officials expect about 625 million lottery tickets to be sold between now and Friday.

If lottery officials are correct there is a 96% chance that someone will pick the winning numbers on Friday, and the expected number of winners, conditional on the jackpot being awarded, is about 3. This means that even after splitting the jackpot, the expected value of a Mega Millions ticket is about 90 cents (including 18 cents in expected consolation prizes).

In general state sponsored lottery games have low expected values for players, relative to other forms of gaming. Lotteries pay out about half of their proceeds in prizes, on average. The rollover nature of the jackpot in state lotteries means that a lottery ticket will have a high expected value on occasion. Until the recent surge in sales, this week would have been the highest expected value for a Mega Millions ticket in the history of the game.

Finally, because the jackpot will be taxed at a fairly high rate a Mega Millions ticket is not a good investment, as Ben Casselman of the Wall Street Journal has noted. After taxes the expected value of a Mega Millions ticket is now much less than $1, unless the top marginal tax rate falls substantially over the next twenty years, which is doubtful. However most gambles with long odds, such as a super trifecta bet at the racetrack, have a pre-tax expected value that is lower than the cost of the wager. So the Mega Millions is an unusually good bet for people who enjoy gambling small amounts at long odds, but it would be a bad investment.