Mistakes: For a Second, I Thought I Could Beat the Lottery.

People make mistakes.  I want to tell you about a mistake I made when I was just figuring out how AP works…  Call it stupidity or ignorance if you want, but I call it a learning experience.

I decided I was going to beat the lottery.  Doh!

I did not aim at the big national games, but smaller “little lotteries” run by individual states.  This was not a good idea, but there was some value in the exercise. 

Now I know there are a few stories out there about people beating the lottery.  Here is a good one:  https://highline.huffingtonpost.com/articles/en/lotto-winners/.  This story is well written and entertaining.  If you are reading this blog, you probably need to read something that is well written and entertaining every once in a while.  That aside, I was not this guy.

Here is my story:

I decided to do a lottery analysis, and when it went +EV, I would jump on it and eventually win.

The example I will use to show you my process is the “Lucky Day” game from the Illinois Lottery.  Here are the rules:

-          Each game played costs $1.00.

-          Pick 5 numbers between 1 – 45

-          2 matches pays $1

-          3 matches pays $15

-          4 matches pays $200

-          5 matches pays the jackpot

-          The jackpot will carry over and grow if no one wins.

That is enough for me to get started.  The first thing I did was to make a spreadsheet to calculate the EV of the game.  The key formula is probability* payback = EV.

The lottery pages like to give frequencies, usually they look like this:

I wanted to double-check their calculations.  To do this, I need something called a combination function.  A combination function is a way of counting things when you choose K things from pool of N things where order of the things does not matter.  The easiest way to calculate a combination is with the “=COMBIN (n, k)” function in Excel.  Another way would be using the “nCr” button on a scientific calculator.  I have no idea why calculators use n & r and excel uses n & k.   It’s probably the same reason one slot machine has an 88% RTP and the next one has 92% RTP. 

I’m using excel, because that’s what I do.

Having a machine spit out a number is awesome, but I think there is value in understanding where that number came from.  It gives me context…  Because of that, I’ll work through a couple calculations by hand using nothing more than the standard calculator that comes with my phone and some grade school math.  As some point 5th grade math hits its limit.  Anyway, this isn’t a math class, so hopefully everything is put together in a way so that most can follow along.  Please be aware (cautioned) there are some long decimals that are rounded, so if you are calculating at home, we could be just a little out of balance.

Let’s talk about the jackpot probability first.  As a reminder, we need to correctly pick 5 numbers out of 5 and there are 45 total choices.

-          Probability of choosing the first correct ball = 5/45 = 0.11111111

-          Probability of choosing the second correct ball = 4/44 = 0.090909091

-          Probability of choosing the third correct ball = 3/43 = 0.069767442

-          Probability of choosing the fourth correct ball = 2/42 = 0.047619048

-          Probability of choosing the fifth correct ball = 1/41 = 0.024390244

I have to get the first ball & the second ball & the third ball & fourth ball & the fifth ball. 

The word “and” means multiplication.  If I multiply those numbers, I get .a probability of .000000818492026659922 for hitting the top line prize.

1/.000000818492026659922 = a frequency of 1,221,759. (I love the 1/x button on my calculator)

There is only 1 way to pick 5 numbers and get 5 numbers correct. That means the web site is right, perfect actually!  1 in 1,221,759 is the frequency.

This is the hard way of doing it.It would be easier to use the combination formula in excel, which works like this:

Would you rather have the probability?  I would.  (Did I mention I love the 1/x button?)  Just do this:

Tools make work easy.

Let’s do one more calculation because it gets a little more complex when you have more ways of making a winner.  Here is how I approached getting the probability of catching 4 of 5 numbers.  The first 4 bullets are the same as the above, but the last bullet changes…

-          Probability of choosing the first correct ball = 5/45 = 0.11111111…

-          Probability of choosing the second correct ball = 4/44 = 0.090909091

-          Probability of choosing the third correct ball = 3/43 = 0.069767442

-          Probability of choosing the fourth correct ball = 2/42 = 0.047619048

-          Probability of choosing the fifth ball incorrectly = 40/41 = 0.975609756

If you multiply those numbers together, you get 0.0000327396810663969, but that is not the probability of hitting the second prize, because there are 5 ways to hit 4 winning balls in this game. 

-          1,2,3,4 | 1,2,4,5 | 1,3,4,5 | 1,2,3,5| 2,3,4,5

Don’t want to do that manually?  Excel to the rescue:

0.0000327396810663969*5 ways = 0.000163698405331984 probability of getting 4 of 5 balls correct.

1/0.000163698405331984 = 6,108.795 frequency.  The web site is close but not exact.  This is why I like checking the math. 

I’ll spare you the rest of the calculations.  Here is the return multiplied by the probability for this game at a break even state.

Here is where things went south.

-          I did not have $1.2M lying around to buy every combination of numbers.

-          I could not figure out a move to get that many tickets printed in a day for the drawing.

-          I’ll spare you the calculations, but multiple winners and taxation was a problem.

With all that said, I never did jump on this “play” if you even want to call it that. 

The reason I took the time to write all this up is that this was the source of a few key realizations.

1)      There is an EV break point between positive and negative.

2)      The EV break point can be calculated.

3)      The shorter the frequency (or the higher the probability) the fewer pulls I need to turn a theoretical advantage into cash in my pocket.

4)      Knowing how to create the last table is very transferable other games.

The only other benefit I hope to get from this is that my wife likes to buy an occasional lottery ticket or two.  I’ve had little success getting her to stop via conversation.  Maybe a stack of words and numbers will help?   Probably not…  Somewhat related, I had another great idea about putting speakers in my dining room.  That also was a total fail.  However, unlike my lottery analysis, I learned absolutely nothing from the dining room exercise.  :-)