Poster ID 19 Binomial vs. Hypergeometric Yvanny Chang What s the difference?

Transcription

1 Poster ID 19 Binomial vs. Hypergeometric Yvanny Chang What s the difference? 1

2 Project Introduction In Statistics, Binomial is replacement and Hypergeometric is no replacement. Can we find examples of them in everyday life? Can we use a Java program to show the difference? Y E S 2

3 Case Study: Powerball Lotto Jan created largest Jackpot in U.S. history of $1.5 billion. Hypergeometric (No replacement) 5 White Balls 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) 3

4 Rule of Red Ball 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? 4

5 Hand Calculation Mega number replacement Mega number without replacement 1.12X difference 5

6 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 Y=X 4+1: Match 4 white balls and 1 red ball 4+0: Match 4 white balls and no red ball Y=X-5 6

7 Java Flowchart Develop a Java program to calculate the probabilities for different X, Y and matching scenarios. This is the Java Flowchart: X Y More scenarios and variables can be simulated in Java without much modification. 7

8 Create Java Script public static double FiveAndOne(int X, int Y){ // Match return combination(x,5)*y; } public static double FourAndZero(int X, int Y){ // Match 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

9 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

10 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

11 Duplicate Powerball probability by JAVA = 292,201, = 11,688, = = = = = = JAVA match all numbers Reference from the Powerball Binomial (replacement) 11

12 Java Program Output Results 12

13 Odd Ratio of Binomial vs. Hypergeometric? With larger pool size, Hypergeometric will approach Binomial (Ratio => 1) Ratio of Matching Binomial Hypergeometric 9%-20% higher chance for Hypergeometric 13

14 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

15 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