Lottery Odds – Are They Ever In Your Favor?

Editor’s Note: This post was previously published –  With the Powerball jackpot reaching $550million for Saturday’s drawing, what better time than now to refresh your memory on what the lottery odds actually look like…

This week’s Powerball jackpot is set to be a record $425,000,000. This prize is paid as an annuity over 29 years. The cash value is estimated to be $278,300,000 if the lucky winner decides to take a lump sum payment.

We all know that the lottery is a sucker’s game, “A tax on people who suck at math.” as I’ve often heard it put. But it seems every few months the Powerball or Mega Millions lottery jackpots reach these astronomical amounts, driving even the most hardened skeptics (read: me.) to run out and buy a ticket.

When the lotto jackpots reach these gigantic numbers the question I always wonder is, “How big does the jackpot need to be for it to be a good decision to buy a ticket?” 

To put it another way - does a lottery ticket ever have a positive expected value (+EV)?

EV Defined

Expected Value is simply the average amount you can expect to gain or lose on an event given the average results of each possible outcome.

To calculate Expected Value (EV) we use the following formula: E[X]= x1p1 + x2p2 + x3p3 +… + xnpn

For those who aren’t mathematically inclined, X is a random variable that can have values x1, x2… with the corresponding probabilities of p1, p2 and so forth.

Say you go to a casino and want to bet $1 on a single number at the roulette wheel. The roulette wheel has 38 spaces and the casino pays out 36:1 when your number hits. So if the ball lands on your number you win $35, if it lands in any of the 37 other numbers, you lose your $1 bet. The EV of that $1 bet would be:

E[$1 bet] = $-1 * 37/38 + $35 * 1/38 = $-0.053

So you are losing just over 5 cents every time the dealer spins the roulette wheel with your $1 on your “lucky” number. Vegas, baby!

Now back to the Powerball jackpot…

Powerball Odds

A Powerball ticket costs $2 and the Powerball website lists the odds of winning the jackpot at 1 in 175,223,510. However, there are also smaller prizes available for hitting fewer numbers. The odds and expected return for the different prizes looks like this:

Powerball Odds and Payout – Cash Option of $278.3 million (before taxes)

The sum of the probability column is 3.14%,  meaning that approximately 1 out of every 31 tickets will win a prize of any kind. The total of the return column, 1.95, means that for every $1 risked in the $278 million jackpot you can expect to get back $1.95Earning 95 cents on every dollar wagered in the Powerball drawing is a really good outcome. However, as you might be able to guess. These numbers are too good to be true.

For starters, we’re completely ignoring the fact that there can be multiple people with the same numbers, thus causing the winners to split the jackpot if their numbers hit. As the jackpot grows, more and more people buy tickets which increases the likelihood of a split pot.

Secondly, and most importantly, this example completely ignores taxes! Applying the top federal income tax rate of 35% to just the top two prizes decreases the return to 79 cents on the dollar. Even with the largest Powerball jackpot in history, federal taxes alone cause the lotto player to have a negative expected return on his/her bet.

Conclusion

Even with record jackpots it’s safe to assume that the odds of turning a profit on your Powerball or Mega Millions ticket won’t be in your favor.

From a purely mathematical standpoint, the lottery isn’t worth playing. But on the rare occasions I play the lottery, I don’t play because it’s a +EV move for my money. I play in case the Sun, the Moon and the stars align and my numbers hit that $278 million prize.

I don’t play the lottery with money I can’t afford to lose, and I don’t expect to see a positive (or any) return on that money. I just dream about what I’d do with all that money, and hope that maybe it’s my lucky day.

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14 Responses to Lottery Odds – Are They Ever In Your Favor?

  1. I will play the lottery when it gets to a high amount, like the current Powerball jackpot. I know the odds are horrible but buying a ticket at least gives you a slight chance. That said, your odds are still terrible even if you buy 10 or 100 tickets, which is why I only buy one. You have an ever so slight chance regardless, and buying one ticket is the least costly approach to still keep you in the game.

    • Jay says:

      Agreed. I only spend a few dollars when I do play. People shouldn’t think that buying $50 in tickets gives them a noticeable advantage over someone who spends $5.

  2. We play the lottery one line per week. Mates of ours won a few million on their first go at it but not everyone is that lucky.

    • Jay says:

      wow! Any idea what they did with the money? Hopefully they were smart with it, but there are a lot of stories of people squandering all their winnings!

  3. I never found gambling enjoyable – to me it is just too stressful and I’d rather take a dollar bill and tear it into a pile of tiny pieces than spend it on a lottery ticket.

    • Jay says:

      That’s not a bad attitude to have! Far too many people get swept up in playing the lottery, especially those scratch off tickets. When in reality tearing a dollar bill in to pieces may offer just as much fun with the same odds of winning! :)

  4. We have a National Lottery in the UK which quotes a probability of 1 in 14 million or so of winning (you have to match 6 numbers – I think it is the same layout as the Canadian lotters). Half of the income goes to good causes, 5% is tax and the rest goes a prizes so the roundness of the book is at best 45% so it is a no-brainer not to ‘play’.

    The point however is the dream so if you do buy a ticket, do it early in the week. Then you have the whole week to dream!

    But your point about sharing winnings is important. In one of the earlier weeks of the National Lottery, 131 people won! Their take-home was less than the many fewer people who matched only 5 numbers!

  5. They claim a portion of each lottery ticket sold goes to good causes (at least in New York) but how much actually does is anyone’s guess.

    Two people split this most recent jackpot, which was over $500 million. Sadly, my ticket was not one of the winners!

  6. Pingback: Worth Mentioning #16 | Planting Our Pennies

  7. I like the methodology you applied above. Now, having said that, I am not one to buy lottery tickets. Frankly, I don’t think it’s bad to one here and there if one truly views it as entertainment. But not because of some belief that winning will happen.

    • Agreed. I rarely play myself, but there’s nothing wrong with buying the occasional ticket as long as you have no dilusions of winning in order to pay off all your debts!

  8. Joe says:

    The lottery is sometimes in your favor by a lot. Example. Missouri lottery is currently running a promotion called “the 12 days of keno bonus promotion”. With that promotion they doubled the payout for 4 spot prizes. For every dollar you spend on average you should get back $1.22. I’ve bought over $10,000 worth of tickets during this promotion and have a profit of about $2,050.

    Yes, MOST of the time the lottery is a tax for people bad a math. But sometimes it’s an all you can eat profit machine for people who are GOOD at math.

  9. steven iwinski says:

    its simple if everyone joined forces and stopped buying lotto tickets the people us will be heard. government makes billions off taxes and i belive they fake winners.all the time tokeep profits some will win most do not exist… government has making money off the people to a science between taxes and child support i cant even afford to have a apt of my own 29 forced to live at home with parents jobs pay shit and there is simply to many people in our country that shouldnt be theycan illegaly come here but legally work because they payinto social security theyll never see most of us wont see anyhow

  10. I just think they are a waste of money and effort. You see so many people spending 10-20$ per week but have not savings. Why not make your own future instead of depending on the odds. They do help the states make a lot of money though.

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