2. How to play
Player approaches a table. He pays a hundred dollars to play the
game. Out of a standard 52 card deck, the casino is assigned 13 hearts,
and the player 13 spades. Each suite is shuffled, and have a random
permutation of thirteen cards. A dealer (casino) draws one card from
the hearts, and the player one card from the spades. If the players card
is higher than the dealers card by face value, the player wins 200
dollars from the game (profit of one hundred dollars). If the players
card is lower than the dealers, OR if there is a tie, the player wins
nothing. If both the player and the dealer draws an ace, the player gets
400 dollars (profit of 300 dollars). Cards are reshuffled after every
game.
3. Theoretical Statistics
• 13 cards for both player and casino
• 169 combinations of cards drawn
• 90 of the combinations result in a player net loss of 100 dollars
• 78 of the combinations result in a player net gain of 100 dollars
• 1 of the combinations result in a player net gain of 300 dollars.
• The expected mean gain for the casino is 3.55 dollars per game.
• x1p1+x2p2+x3p3…=μ
• -100(90/169)+100(78/169)+300(1/169)=3.55
4. Theoretical Statistics continued
• The expected standard deviation is 48.78
• σ2x=Σpi(xi-μx)2
• ((90/169)(-100-3.55))2+((78/169)(100-3.55))2+((1/169)(300-3.55))2=2379.5532
• √(2379.5532)=σ=48.78
5. Experimental Statistics
•
•
•
•
We simulated 30 trials
The player lost 18 times for a net loss of 300 dollars
The casino won 18 times for a net gain of 300 dollars
The experimental mean for the casino was +10 dollars per game
• Net gain/# of trials=mean gain for casino per game
• 300/30=10
• The experimental standard deviation for the casino is $99.65
6. Conclusion
On average, the casino wins $3.55 per game with a standard
deviation of $48.78. With a small 30 trials, our mean was $10 with a
standard deviation of $99.65. As we simulate more and more trials, we
will come closer and closer to the theoretical mean and standard
deviation. A potential improvement in out game is that we can make
different options for what’s paid to play and respective winnings (ex,
pay 20 dollars per play, win or lose 20, get 80 for a double ace).
