THE GAME OF CRAPS
In craps a player rolls 2 dice. If she gets a 7 or an 11, she wins at once whatever she has bet.
If he throws a 2,3 or 12, she loses at once. If she initially rolls a 4,5,6,8,9 or 10, she keeps
rolling the dice until she either wins by rolling the same number he rolled initially ("makes his point")
or loses by rolling a 7 ("craps out").
If we look in on the game then there are nine different situations or states that we might see.
1. The game is about to begin. The player has not yet thrown the dice for the first time.
2. The player has won.
She threw a 7 or 11 on the first throw ( P = _____ ) or made her point on a subsequent throw.
3. The player has lost.
She threw a 2 or 3 or 12 on the first throw ( P = ______ )or a 7 on a subsequent throw.
4. The player threw a 4 on the first throw (P = ______ ) and is ready to make a subsequent throw.
Her "point" is 4.
5. The player threw a 5 on the first throw (P = ______) and is ready to make a subsequent throw.
Her "point" is 5.
6. The player threw a 6 on the first throw (P= _______) and is ready to make a subsequent throw. Her "point" is 6.
7. The player threw a 8 on the first throw (P = _______) and is ready to make a subsequent throw. Her "point" is 8.
8. The player threw a 9 on the first throw (P= ________)and is ready to make a subsequent throw. Her "point" is 9.
9. The player threw a 10 on the first throw (P = ______) and is ready to make a subsequent throw. Her "point" is 10.
The initial probability matrix is : [ ___ , ___, ___, ___, ___, ___, ___, ___, ___ ]
The transition matrix is :
TO
1 2 3 4 5 6 7 8 9
1 ____ ____ ____ ____ ____ ____ ____ ____ ____
2 ____ ____ ____ ____ ____ ____ ____ ____ ____
3 ____ ____ ____ ____ ____ ____ ____ ____ ____
4 ____ ____ ____ ____ ____ ____ ____ ____ ____
FROM
5 ____ ____ ____ ____ ____ ____ ____ ____ ____
6 ____ ____ ____ ____ ____ ____ ____ ____ ____
7 ____ ____ ____ ____ ____ ____ ____ ____ ____
8 ____ ____ ____ ____ ____ ____ ____ ____ ____
9 ____ ____ ____ ____ ____ ____ ____ ____ ____
We will use the matrix capabilites of the TI-83 to determine the long-range probabilities of winning and losing.
¨ open the TI-83 Emulator
¨ press 2nd and then the MATRX key (above the x-1).
¨ move over to the EDIT menu and press ENTER.
¨ for matrix A change the dimension to 1 by 9 to represent the initial probability matrix and enter the values from this sheet.
¨ when done press 2nd and QUIT (above MODE)
¨ press 2nd and MATRX, move over to EDIT and down to highlight matrix B
¨ change the dimension of matrix B to 9 by 9 to represent the transition matrix and enter the values from this sheet. When done, press 2nd and QUIT.
¨ To calculate the long term probabilities, do the following :
¨ press 2nd, then MATRX
¨ press 1 to select matrix A
¨ press the multiply key
¨ press 2nd, then MATRX, then press 2 to select matrix B
¨ press the exponent key (^) below the CLEAR key
¨ enter an exponent of 25 and press the ENTER key
Write the resulting 1 by 9 matrix here :
[ ______ , ______ , ______ , ______ , ______ , ______ , ______ , ______ , ______ ]
In the long term you are likely to win _______ % of the time, lose _______ % of the time. GO BACK