Poster ID 19

Binomial vs. Hypergeometric

Yvanny Chang

What’s the difference?

1 © IEOM Society International

Project Introduction

Can we find examples of them in

everyday life?

Y E S

In Statistics, Binomial is replacement and Hypergeometric is no

replacement.

Can we use a Java program to

show the difference?

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Case Study: Powerball Lotto • Jan. 13 2016 created largest Jackpot in U.S. history of $1.5 billion.

• Binomial (Replacement)

–1 Red Ball (can be the same

number as the 5 White Balls)

Question: what if red ball has no duplicate?

(Hypergeometric; No Replacement)

• Hypergeometric (No replacement)

–5 White Balls

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47

White Balls

(No replacement, or

no duplicates)

Red Balls

(Replacement

(duplicates allowed))

What do you think the probability difference between binomial

and hypergeometric is?

Rule of Red Ball

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Hand Calculation

1.12X difference

Mega number replacement Mega number without replacement

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General Java Model

For X number of white balls, Y number of red balls, what is winning

chance for the following cases?

5+1: Match 5 white balls and 1 red ball

5+0: Match 5 white balls and no red ball

4+1: Match 4 white balls and 1 red ball

4+0: Match 4 white balls and no red ball

Y=X

Y=X-5

6 © IEOM Society International

Java Flowchart

X Y

Develop a Java program to calculate the probabilities for different X, Y

and matching scenarios. This is the Java Flowchart:

More scenarios and variables can be

simulated in Java without much modification. 7 © IEOM Society International

Create Java Script

public static double FiveAndOne(int X, int Y){ // Match “5 + 1”

return combination(X,5)*Y;

}

public static double FourAndZero(int X, int Y){ // Match “4 + 0”

return combination(X,5)*Y/combination(5,4)/combination(X-5,1)/combination(Y-1,1);

}

public static void main{ // Main

program to print out the result

int X=59, Y=59;

System.out.println("5 + 1 X="+X+" Y="+Y+": Binomial is " + FiveAndOne(X,Y));

System.out.println("5 + 1 X="+X+" Y="+Y+": Hypergeometric is " +FiveAndOne(X,Y-5));

System.out.println("5 + 1 X="+X+" Y="+Y+": Ratio is "

+FiveAndOne(X,Y)/FiveAndOne(X,Y-5));

}

8 © IEOM Society International

Matching Jackpot 5+1 Probability For the scenario of Matching 5 in X number of white balls, 1 in Y

number of Red balls.

● The total space= C (X, 5) * C (Y, 1)

● The probability of matching 5 White Balls = C (5,5)= 1

● The probability of matching Red Ball= C (1, 1)= 1

● The total probability of 5+1 is 1 in C (X, 5) * C (Y, 1)

C (5, 5)

C (1, 1) 9 © IEOM Society International

Matching 4+0 Prize Probability

For the scenario of Matching 4 in X number of white balls, 0 in Y

number of Red balls.

● The total space= C(X,5)*C(Y,1)

● The probability of matching 5 White Balls = C(5, 4) * C(X-5,1)

● The probability of matching Red Ball= C (Y-1, 1)

● The total probability of 4+0 is 1 in C(X,5)*Y/(C(5,4)*C(Y-5,1))/(Y-1)

C (5, 4)* C(X-5,1)

C (Y-1, 1)

10 © IEOM Society International

Duplicate Powerball probability by JAVA

5 + 1 = 292,201,338

5 + 0 = 11,688,053.52

4 + 1 = 913129.18

4 + 0 = 36525.17

3 + 1 = 14494.11

3 + 0 = 579.76

2 + 1 = 701.33

1 + 1 = 91.98

Reference from the Powerball Binomial (replacement)

http://www.lottonumbers.com/lotto-odds-calculator.asp

JAVA

match all

numbers

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Java Program Output Results

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Odd Ratio of Binomial vs. Hypergeometric?

Ratio of Matching

Binomial

Hypergeometric

9%-20% higher chance for

Hypergeometric

With larger pool size,

Hypergeometric will

approach Binomial (Ratio

=> 1)

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Results and Conclusions

● This Java project shows the differences between Binomial

(Replacement) and Hypergeometric (No replacement) in statistics

and probability.

● Using Java program, we can easily test the probability for different

scenarios and variables.

● The result matches with Powerball probability.

● Results also shows that Binomial is 9%-20% higher chance for

Hypergeometric.

● With larger pool size, Hypergeometric will approach Binomial (Ratio

=> 1).

14 © IEOM Society International

Future works

• Account for more than one Jackpot Winner. Current

probability calculations only work if there is only one

jackpot winner

• Calculate for other lotteries (Super Lotto…). See if

other lotteries have different or similar results.

15 © IEOM Society International