Adventures in Lottery Playing - What Was The Expected Value of Mega Millions Tickets?

April 5th, 2012 by 

Editor's note: Did we screw up the math? Of course not, this isn't amateur hour! However, we did use the wrong estimate for ticket sales to compute our odds. We used a number north of 1.4 billion, when the true number, 1.49 billion, counted all of the previous drawings which hadn't yet declared a winner. The actual number wa sin the mid 600 millions. See our nifty Mega Millions Odds Calculator to find the real value - around 59 cents per ticket we purchased. Better? Sure. Still stupid.

Happy Thursday, readers!  I'm delighted to share with you this short story about my entry into the recent Mega Millions drawing.  Alas, I didn't win anything, but I'll share this little anecdote with you nonetheless.  Skip to the end if you are easily bored by math!

Let's set the stage: Mrs. DQYDJ had come home from the market on the day of the drawing in a bad mood.  That mood worsened when she dropped a glass bottle of apple juice onto our garage floor.  Dissuading her from crying over spilled juice, I recommended that she take sixty dollars down to a local convenience store and enter us into the heavily hyped Mega Millions drawing.  Not being a gambling couple, she was initially surprised, but when I explained that the expected value of a ticket wasn't that horrible, she acquiesced.  We ended up with 15 manual entries into the Mega Millions drawing, and another 45 quick picks.

What a Waste of $60!

...but was it really?  I mean, did the lead editor of DQYDJ really pay $60 to prevent $3.50 in apple juice from becoming a big thing?  Well, I did, but since we've been rambing to your guys about the lottery and gambling in the last few weeks, let's discuss the expected value of my ticket.

A brief note, California pays Mega Millions prizes on a pari-mutuel basis, so their prizes depend on ticket sales.  The main result of this?  Slightly different expected values.  Since 9 out of 10 Americans don't live in California (and 100% of non-Americans), I will calculate the odds based on the official published values of the drawing.

Naively, you might say that I had a positive expected value.  In the Mega Millions, there are 5 mutually exclusive regular numbers from 1 to 56 and a 'MEGA' number from 1 to 46.  The odds of a correct choice works out to 1 in 175,711,536 times my 60 tickets to win $462,000,000... or $2.63 cents in expected value per $1 ticket.  Even factoring in tax, which I'm going to estimate at 45% for most of the country after a big win, and each ticket would still have an expected value of $1.44 based on the main prize alone.

Of course, that isn't the case.  First, a positive - there are a number of fixed prized (for non-Californians, but the prizes were similar in CA) from $250,000 down to two.  To find the expected value, divide the prize by the odds (1 in x).  I get 18.189 cents per ticket in added EV, which becomes 10 cents after I apply the 45% tax rate (you might argue that the smaller prizes might not cause someone to pay the higher tax - maybe so.  Feel free to respin it, but the numbers will be close.)

So, With an EV of $1.54, Did You Buy $92.40 With Your $60?

Not winning numbers for the March 30, 2012 Drawing of Mega Millions

Not quite, pals, but thanks for assuming my buy was completely rational!  From all I can tell, Mrs. DQYDJ and I weren't the only people to buy tickets for the March 30 drawing.  In fact, $1.46 billion of tickets were estimated (can't find any newer numbers).  One site dedicated to calculating the odds of lottery winnings suggests using a poisson distribution and assuming a random distribution of tickets - which is close enough for our purposes (for a discussion of why this won't work with manual picks, see this article explaining the effect of manual lottery picks).  This makes our input to the poisson, our mean value = 1.46B/175,711,536, or 8.309073116292148.  (For the more wonkish out there, as the mean value to poisson rises it begins to approach a binomial distribution).

Working out the Poisson for 0 through 16, we find that the expected value of the jackpot is just $64,709,583.27, a far cry from $462,000,000 (the most common outcome, at 13.9% would have been an 8 way tie.  In real life, it looks like there were 3 winners, which had odds of 2.35%.  By the way, there was a .246% chance of no winners!).

OK, Fess Up.  How Much Money Did You Waste?

$64,709,583.27 after that 45% tax?  $35,590,270.80.  Your actual EV based upon the grand prize was 20.255 cents.  Add back that 10 cents for the smaller prizes, and every dollar you spent on the lottery bought you just 30.255 cents in expected value.

Our little adventure in lottery ticket buying cost us $60 x (1-.30255) = $41.85.

So yeah, $41.85 over a little spilled juice.  Whoops...

Are my assumptions fair?  Did you buy any tickets for the Mega Millions?  Have you ever obsessively calculated the expected value of a ticket like I just did?



PK started DQYDJ in 2009 to research and discuss finance and investing and help answer financial questions. He's expanded DQYDJ to build visualizations, calculators, and interactive tools.

PK lives in New Hampshire with his wife, kids, and dog.

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