Pick 3 Ohio Lottery – A Matter of Randomness
Arun Pidugu
High School Senior, Pennsylvania
And
M B Rao
University of Cincinnati
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A preamble
British experience:
Casinos use dice and other artifacts supposedly random in a variety of games they
offer. Are the dice used really honest? Are the artifacts really random? Are slot
machines honest?
Checking the honesty of dice, for example, is government’s job. Someone from
the ministry in charge of casinos does the job. He checks whether or not the dice
are loaded. How?
American experience:
There is no governmental interference. Casinos are trusted? One can buy crooked
and loaded dice from the internet. There are self-help manuals available how to
load dice.
Generally, lotteries are state-sponsored. Lotteries do pick numbers supposedly
random from a prescribed set of numbers. There is no reason to suspect that the
process is not random. There is no harm in checking whether the process is
random. It will be a useful experience to learn some of the mores and ways of
statistics. Fortunately, we do get data on the chosen numbers from the state
agencies.
Ohio Pick 3 Lottery
A 3-digit number is chosen twice daily. There are 1000 possibilities from 000 to
999.
Bet ‘straight.’ Bet on a specific three-digit number. Bet one dollar. If the number
shows up, you win $500.
Your Winnings:
-1
499
Pr:
0.999
0.001
E(Your Winnings) = -1*0.999 + 499*0.001 = - 0.5
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Another interpretation: For every dollar bet, the state makes 50 cents on average.
Other variations:
Bet on ‘6-way Box.’ Choose three distinct single-digit numbers, say 1, 2, and 3.
You win if 123, 132, 213, 231, 312, or 321 show up. For every dollar bet, you get $
83 if you win.
Your Winnings:
-1
82
Pr:
0.994
0.006
E(Your Winnings) = -0.994 + 82*0.006 = -0.502
Bet on ‘3-way Box.’ Choose three single-digit numbers, two of which are the
same. For example: 0, 2, and 2
You win if 022, 220, or 202 show up. You get $167 if you win.
Your Winnings:
-1
166
Pr:
0.997
0.003
E(Your Winnings) = -0.997 + 166*0.003 = -0.499
Let us go to their website …
Data:
They have an archive in which the results of the drawings are stored from 1994.
The data are accessible.
Question: Are the numbers drawn at random?
Methodology
We have 20 years data starting from 1994 to 2013. For each year we have the
complete list of numbers chosen. It is about 730 3-digit numbers. We summarize
the data. Let
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X0 = Number of 3-digit numbers not chosen
X1 = Number of 3-digit numbers chosen exactly once.
X2 = Number of 3-digit numbers chosen exactly twice.
X3 = Number of 3-digit numbers chosen exactly thrice.
And so on.
X0 + X1 + X2 + … = 1000
1994
X0 = 731
X1 = 233
X2 = 29
X3 = 6
X4 = 1
Sum = 1000
If the numbers were picked randomly, we can calculate expected number of Xi s.
E(X0) = 731.1
E(X1) = 229.1
E(X2) = 35.8
E(X3) = 3.7
E(X4) = 0.3
Is there a good agreement between the observed and expected under the null
hypothesis of randomness?
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Calculate the chi-squared statistic χ2 = ∑
(
)
and apply the chi-squared statistic.
The observed value of the chi-squared statistic is 3.61 with 3 degrees of freedom.
The evidence is not strong enough to reject the null hypothesis of randomness.
Mathematics
We use the methods of Urn Models.
A theoretical model of randomness:
Create 1000 urns one for each 3-digit number.
Make 730 marbles of same size one for each pick-3.
Throw each marble at random into one of the urns. The chances that an urn gets
the marble are 1/1000.
Let
Y1 = Number of marbles Urn 1 (000) gets;
Y2 = Number of marbles Urn 2 (001) gets;
And so on.
The random vector (Y1, Y2, Y1000) has a multinomial distribution with N = 730
and pi = 1/1000, i = 1 to 1000.
Occupancy numbers:
X0 = Number of urns with 0 marbles
X1 = Number of urns with 1 marble
And so on.
E(Xi) = 1000*(
) (
)
(
)
, i = 0, 1, 2, …
Some computational sophistication and additional probability theory is needed to
calculate expectations.
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Conclusions:
1. For every year under examination, the chi-squared statistic value is very
low.
2. Randomness in pick-3 selections gets a clean bill of health.
Future work:
1. Look at Pick-4 and Pick-5 selections. Evaluate which one is better in terms
of expected winnings from the bettor’s point of view.
2. How much money Ohio makes on these? Can we get data on the amount
bet on Pick-3 year by year?
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