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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?
2 © IEOM Society International
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
3 © IEOM Society International
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
4 © IEOM Society International
Hand Calculation
1.12X difference
Mega number replacement Mega number without replacement
5 © IEOM Society International
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
11 © IEOM Society International
Java Program Output Results
12 © IEOM Society International
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)
13 © IEOM Society International
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
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