The document describes a gumball game where players pay $0.75 to receive a gumball of a random color that corresponds to a monetary value they can win. Black gumballs yield $0, while rarer colors like blue and ones with the owner's face yield higher values. Statistics show the theoretical expected profit per game is -$0.15 for the player and $0.15 for the owner, but simulations estimate -$0.41 for the player and $0.41 for the owner. The owner is likely to profit from this game.

2. The Gumball Game

3. Description of the Game
• Pay $0.75 to get a gumball. The different colors of the
gumballs symbolize values of money that you can win.
Black is $0, yellow is $0.25, green is $0.75, red is $1.00,
blue is $5.00, and Mr. Evans’ face is $25.00. The player
wins the amount of money that correlates with the color (or
face).
• There will be 70 black balls, 10 yellow balls, 9 green balls,
6 red balls, 4 blue balls, and 1 ball of Mr. Evans’ face.
• After each game (the purchase of one gumball) the same
exact colored gumball that comes out will be replaced.

4. The Stats
Colors of
Gumballs
Black Yellow Green Red Blue Mr.
Evans’
Face
X 0 .25 .75 1 5 25
P(X) .7 .1 .09 .06 .04 .01
•Theoretical Expected
Value=(0*.7)+(.25*.10)+(.75*.09)+(1*.06)+(5*.04)+(25*.01)=$0.60
•Theoretical Standard Deviation=√(.7(0-.60)²+.1(.25-.60)²+.09(.75-
.60)²+.06(1-.60)²+.04(5-.60)²+.01(25-.60)²)=$2.65
•1-70=black; 71-80=yellow; 81-89=green; 90-95=red; 96-99=blue;
100=face
•Randint(0, 99, 50) L1 then assort them into ascending order, assign
values for each number in L2, then use 1VarStats to find mean and
standard deviation
•Simulated Expected Value=$0.34
•Simulated Standard Deviation=$1.00

5. Conclusion
According to the theoretical value, for each round , the player is expected
to win on average $.60, give or take roughly $2.65 (the standard deviation).
However, since the cost of the game is $.75, the player ‘s expected profit is
$.60 - $.75 = $-.15. This means that for each game, the player is expected to
lose $.15, and therefore the company is expected to gain a profit of $.15 for
every $.75 put into the machine. According to the simulation expected mean,
the player is expected to win about $.34, for each game, and therefore the
player’s expected profit is &.34 - $.75 (cost to play) = $-.41. As a result, the
company is expected to make $.41 profit for each round, according to the
simulation, give or take only $1.00, as compared to the theoretical standard
deviation of $2.65. This expected profit for the company is even larger than the
theoretical expected profit, making it seem like the company definitely has a
good chance of making a profit. As for improvements that could be made to the
game, the amount of each colored gumball could be altered in order to make
the number of black and yellow gumballs (the gumballs that give the company
profit) greater and give the company a better pay-out. Another alteration could
be to increase the amount of money it costs to buy the gumball from $0.75 to
$1.00 since more people carry around dollars instead of change. If the cost was
increased then the game could be changed to increase the player’s expected
profit in order to keep people playing. This could be done by adding another
gumball with Mr. Evans’ face on it or adding gumballs with colors that give the
player a profit.