Team EE

Probability and Chance: A Ping Pong

Ball Game

How to Play:

In this game, the player is given a bucket of twelve white ping pong balls. Each ball has one number from 1-6 written on it. In total, the numbers 1, 2, 3, 4, 5, and 6 each appear two times. After paying the game fee of $4, the player reaches in and grabs two ping pong balls. He is to set the first one down before picking out the next. The absolute value of difference between the two numbers is calculated and used. A difference of 5 earns the player $15. A difference of 4 earns the player $10. A difference of 0, 1, 2, and/or 3 yields no monetary win.

Theoretical Results

• Theoretical Expected Value (Mean): (1/6)(-4)+(5/18)(-4)+(2/9)(-4)+(1/6)(-4)+ (1/9)(6)+(1/18)(11)

= -2.06

• Theoretical Standard Deviation: √[(1/6)(-4-(-2.06))²+(5/18)(-4-(-2.06))²+(2/9)(-4-(-2.06))²+(1/6)(-

4-(-2.06))²+ (1/9)(6-(-2.06))² + (1/18)(11-(-2.06))²]

= 4.45

Difference 0 1 2 3 4 5

Winnings -$4 -$4 -$4 -$4 $6 $11

Probability 1/6 5/18 2/9 1/6 1/9 1/18

Experimental Results

• Experimental Expected Value (Mean):(1/25)(-4)+(11/50)(-4)+(11/25)(-4)+(7/50)(-4)+ (2/25)(6)+(2/25)(11)

= -2.00

• Experimental Standard Deviation:√[(1/25)(-4-(-2))²+(11/50)(-4-(-2))²+(11/25)(-4-(-2))²+(7/50)(-

4- (-2))²+ (2/25)(6-(-2))² + (2/25)(11-(-2))²]

= 4.69

http://youtu.be/ftmCz7OlaEw

Difference 0 1 2 3 4 5

Winnings -$4 -$4 -$4 -$4 $6 $11

Probability 1/25 11/50 11/25 7/50 2/25 2/25

ConclusionBy comparing my theoretical and experimental results, I can

see that my actual results were pretty close to the theoretical results. From the theoretical data, I expected a mean or house advantage of about $2.06 on average per game. After my 50-trial simulation, I observed that the house advantage was extremely similar at about $2.00 on average per game, which is only a 6 cent difference. Both theoretical and experimental standard deviation values were also close to each other; there was only a difference of 0.24 (4.69-4.45) with my actual standard deviation being the larger one. This shows that the overall values’ range I got did not deviate that far away from the expected results’ range. A way I could improve upon this game is by increasing the number of ping pong balls used. Rather than twelve, which is the bare minimum required for the game to work, I could use, for example, 60 or even 120 ping pong balls. Also, I could increase the amount of trials to improve my closeness to my expected values.