Expected Value of Lottery Ticket

In my previous post I have explained how the lottery number system works and forecasting a winning number isn't possible with confidence greater than the random chance. This randomness of the data doesn't help us do any kinda analytic. Only thing that can be done is to verify the lottery prediction doesn't have any skewness towards a particular set of numbers.

Considering randomness involved in the lottery system, still few people found a way to win the lottery with higher confidence level or they are sure that they can make profit out of the lottery. I showed a few such instances in my previous blog entry. In this blog let's dive into one such case more deeply and see how and what helped them exploit the lottery system.

In short they exploited the issues in Game Settings and make use of the Law of Large Numbers to maximize their profit.

The game setting

I'm taking a game named WINFALL played in the 2004 period in Michigan. Michigan State scrapped the previous game with this one to lure people to buy more lottery tickets. WINFALL offered better prize money for lower matching numbers and an option of distributing all prize money if the jackpot hits 5Million or above and nobody won the jackpot for the subsequent draw.

The game setting is:-

  1. Players can select 6 numbers randomly from 1 to 49.
  2. One lottery means this selected 6 numbers and costs $1
  3. The lottery draw happens weekly twice, Wednesday and Saturday.
  4. The Michigan lottery randomly picks 6 numbers as jackpot numbers on a particular drawing day.
  5. Anybody has these exact 6 numbers gets the full jackpot money.
  6. Those who have fewer matches like, 3 matching numbers get 5 dollar, 4 matches get 100 dollar and 5 matches get 2500 dollar.
  7. When the jackpot crosses 5Million, and nobody gets the jackpot then the full jackpot is distributed to all the lower matching lotteries. This means those lower matching numbers get 10x more money. 3 number matches get 50 dollar, 4 number matches gets 1000 dollar and 5 number matches gets 25000 dollar.

The fact was most of the time nobody gets Jackpot, those cases lower matching numbers gets corresponding prize money. Obviously when the jackpot crosses 5Million there will be high demand for tickets, because each 3 or higher match gets 10x more prize unless nobody gets the jackpot on that draw.

At this stage it looks like a standard lottery game played in those times. What you think about the weakness of this game and how people exploited it to win guaranteed profit out of this lottery game. The usual idea was the "lottery is a tax on poor people" and any person who knows the chance of winning won't go near the lottery games, including me :).

But what was the reason math geeks Selbee's and other MIT students see an opportunity here ?. They found a way to increase their chance of winning. See below to know how they cracked it.

Your winning chances

Let's see first what are the visible chances here to win a jackpot or other prizes. The game we are discussing is 6/49, which means pick 6 numbers from 1-49 numbers. We need to know what is the chance of getting a jackpot or what's the chance of getting 3, 4 or 5 matching numbers when a lottery draw happens.

Urn Model

To easily model this, think of we have 49 balls which are labeled 1-49 in a Urn. Each lottery purchase means picking 6 balls from this Urn with replacement. Also think of the six numbers which are going to be picked for jackpot are colored in BLUE and all other balls are RED in color.

So when you buy a lottery ticket, what are the chances of getting blue balls ? Getting 3 or more blue balls means corresponding prize waiting for you.

$$ \begin{array}{l} Odds\ of\ getting\ no\ blue\ ball\ at\ all\ \ \ \ \ \ \ \ =\ \frac{^{6} C_{0}{}{}{} \times ^{43} C_{6}}{^{49} C_{6}} \ =\frac{6096454}{13983816} \ =\ 0.435\\ \\ Odds\ of\ getting\ 1\ blue\ ball\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{^{6} C_{1}{}{}{} \times ^{43} C_{5}}{^{49} C_{6}} =\ \frac{5775588}{13983816} \ =\ 0.41\\ \\ Odds\ of\ getting\ 2\ blue\ ball\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{^{6} C_{2}{}{}{} \times ^{43} C_{4}}{^{49} C_{6}} =\ \frac{1851150}{13983816} \ =0.13\\ \\ Odds\ of\ getting\ 3\ blue\ ball\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{^{6} C_{3}{}{}{} \times ^{43} C_{3}}{^{49} C_{6}} \ =\ 0.017\ \\ \\ Odds\ of\ getting\ 4\ blue\ ball\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{^{6} C_{4}{}{}{} \times ^{43} C_{2}}{^{49} C_{6}} \ =0.00096\\ \\ Odds\ of\ getting\ 5\ blue\ ball\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{^{6} C_{5}{}{}{} \times ^{43} C_{1}}{^{49} C_{6}} \ =\ 1.844\ *\ 10^{-5}\\ \\ Odds\ of\ getting\ 6\ blue\ ball\ or\ Jackpot\ \ \ =\frac{^{6} C_{6}{}{}{} \times ^{43} C_{0}}{^{49} C_{6}} \ =\ 7.15\ *\ 10^{-8} \ \ \end{array} $$

With the above chances if we buy a ticket on a normal drawing period, how much return can you expect ?

Matches and winning prize,

Number of Match Prize money($)
3 5
4 100
5 2500
6 2000000

Based on the above table and the chances of winning, calculate the expected prize money for a ticket.

\begin{array}{l} Expected\ Return\ is\ =\ (\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.43\ *\ 0\ +\ 0.41\ *\ 0\ +\ 0.13\ *\ 0\ +\ 0.017\ *\ 5\ +\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.00096\ *\ 100\ +\ 0.000018\ *\ 2500\ +\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.000000071\ *\ 2000000\\ ) \ \ \sim =\mathbf{\ 0.4} \end{array}

That means you get close to 40 cents for a dollar, that means if you play enough number of times with this prize money you lose 60 cents per dollar. Otherwise 60% lose.

Till this it's a well known fact that lottery games are kinda a "tax on poor", always state make money out of it for sure. In the above case, The State makes approximately 60 cents for every ticket they sell.

The loophole !

As we saw in the Game Setting section, when the jackpot size reaches 5Million, the next draw will ensure the entire money will be rolled down to all other lower matching lotteries unless there is a jackpot winner.

The new prize money range, when Jackpot crosses 5 Million and nobody wins the jackpot on the following draw.

Number of Match Prize money($)
3 50
4 1000
5 25000
6 5000000

With this scenario how much do you make when you buy a ticket ? Let's recalculate the new expected value of a ticket again.

\begin{array}{l} New\ Expected\ Return\ is\ =\ (\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.43\ *\ 0\ +\ 0.41\ *\ 0\ +\ 0.13\ *\ 0\ +\ 0.017\ *\ 50\ +\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.00096\ *\ 1000\ +\ 0.000018\ *\ 25000\ +\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.000000071\ *\ 5000000\\ ) \ \ \sim =\ \mathbf{2.615} \end{array}

Now for every dollar you make close to $2.6, ie; You can expect close to 260% profit. That means if you buy tickets when the jackpot reaches 5Million you get a very good return from it.

WAIT, this case, if you buy ONLY one ticket will it ensure a return of 2.6 dollar ? NO !. Why so, that’s where the real meaning of Expected Value comes in. To get close to the Expected Value, you have to sample enough times otherwise buy enough tickets. So overall return of all your tickets comes close to the expected value of 2.6 dollar per ticket.

So what these people have done is they purchased a large number of tickets with the help of their friends and partners. Even a few formed companies (Or lottery syndicate) that give shares if some pool in their cash. That's how players like Selbee's pooled enough money to purchase a lot of lotteries so that the overall return will be close to the expected value mentioned above. The summary is play in bulk and get far better ROI.

There are more different strategies used to maximize the return or ensure how fast we can converge to the expected value. I have attached few references, please read to get more background about it.

Let's Implement it

# prize money when we have n blue balls ? 
prize_money = {0: 0, 1: 0, 2: 0, 3: 5, 4: 100, 5: 2500, 6: 2000000 }

# When jackpot hits 5Millon or above.
#prize_money = {0: 0, 1: 0, 2: 0, 3: 50, 4: 1000, 5: 25000, 6: 5000000 }

# Probability of getting n blue ball ?
expected_win_prize = 0

matches = []
prizes = []
for i in range(0, 7):
    w_prob = (comb(43, 6-i) * comb(6, i)) / comb(49, 6)
    #total+= w_prob
    print(f"Probability of getting {i} blue ball = {w_prob}")
    expected_win_prize += w_prob * prize_money[i]
    matches.append((i, w_prob))
    prizes.append((prize_money[i], w_prob))

print(f"Expected winning prize of a single lottery is : {expected_win_prize}" )

On Normal Drawing day the chances are listed below,

# prize money when we have n blue balls ? 
prize_money = {0: 0, 1: 0, 2: 0, 3: 5, 4: 100, 5: 2500, 6: 2000000 }

Probability of getting 0 blue ball = 0.4359649755116915
Probability of getting 1 blue ball = 0.4130194504847604
Probability of getting 2 blue ball = 0.13237802900152576
Probability of getting 3 blue ball = 0.017650403866870102
Probability of getting 4 blue ball = 0.000968619724401408
Probability of getting 5 blue ball = 1.8449899512407772e-05
Probability of getting 6 blue ball = 7.151123842018516e-08

Expected winning prize of a single lottery is : 0.37426121739588103

On the 5Millon Jackpot day, the chance of winning for a given ticket.

#prize_money = {0: 0, 1: 0, 2: 0, 3: 50, 4: 1000, 5: 25000, 6: 5000000 }

Probability of getting 0 blue ball = 0.4359649755116915
Probability of getting 1 blue ball = 0.4130194504847604
Probability of getting 2 blue ball = 0.13237802900152576
Probability of getting 3 blue ball = 0.017650403866870102
Probability of getting 4 blue ball = 0.000968619724401408
Probability of getting 5 blue ball = 1.8449899512407772e-05
Probability of getting 6 blue ball = 7.151123842018516e-08

Expected winning price of a single lottery is : 2.669943597656033

Checkout how the expected value changed. This was the key insight those people saw in the game and exploited. It's not that straight forward, higher expected value doesn't guarantee a profit of $2.45 for every $1 lottery ticket, that intuition comes only for those who have statistics background.

Expected value really means, by the Law of Large Number if we sample a random variable variable enough number of times the values of random variable will converge to its expected value or sample mean. Here how do we know how many samples to take from this random variable ? , That is only possible if we know full distribution properties of this random variable. Most of the cases that's not possible due to infinite possibilities, we have to stop the sampling somewhere with an acceptable error.

Instead of addressing it full proof, going with a large number of tickets or samples from the random variable is enough to make profit out of this game, that's what these players did. They did some tweaking to maximize their returns from their finite resources or purchasing power.

Here the random variable is a categorical variable with 7 values ( How many lottery numbers matched against the jackpot number ).

How Michigan lottery fixed this problem

The problem was with strictly following the rules of the game, they didn't take it seriously when bulk purchase of lottery happens across multiple stores. This game has an inherent nature like the demand can drive the volume of sale, which is exploited here to make the chances better for players like Selbee's and others.

Main things to control are,

  1. Restricting bulk selling.
  2. Closely monitoring sudden spike in selling volumes.

After the news came out they stopped this game and added more restrictions on the bulk purchases.

But some other States in the USA had similar lottery games and people like Selbee's exploited that chance also, But all of it eventually closed down after sighting these issues.

In either case the State doesn’t lose any money. Usually the State uses the profit from lottery games to do more social development.

Kerala Lottery

Kerala is one of the States in India, where the Government has been running the lottery since 1967. Till very recently a lot of private lotteries were present in the market. Currently the Government is fully controlling the system, no private parties can run a lottery.

From a quick look, above mentioned like mass lottery purchase isn't possible with Kerala Lottery, as here the lotteries are in printed form, and state only prints fixed number of tickets based on the previous selling trends. Also there are special serial numbers for each District in Kerala. Only possible issues are duplicate tickets and other similar fraudulent activities, which are easily identifiable.

Currently I'm picking up the probabilistic programming using pyro and edward, soon I will model this problem in probabilistic programming to validate the same by simulating the lottery game Coming soon.

NOTE: The entry was taken from its original source and cross posted here.

References

  1. http://www.casinocitytimes.com/news/article/michigan-lottery-launches-new-winfall-game-130262
  2. https://www.mass.gov/files/documents/2016/08/vv/lottery-cash-winfall-letter-july-2012.pdf
  3. http://www.jofamericanscience.org/journals/am-sci/0201/06-lihao-0106.pdf
  4. https://en.wikipedia.org/wiki/Lottery_mathematics
  5. https://www.youtube.com/watch?v=U7f8j3mVMbc
  6. https://highline.huffingtonpost.com/articles/en/lotto-winners/
  7. http://www.keralalotery.in/p/kerala-lottery.html
  8. Book - How Not to be Wrong
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