Hi, My name is Grace and I am a parent of a student is 12th grade with the following question in Linear Algebra.
Jack is playing pool with Jim for $1 a game. He has only $2 and decides to play until he goes broke or has $5, at which point he will quit and go out for a pizza with Jim(Dutch treat). Jack knows from past experience that he beats Jim 60% of the time. What is the probability that Jack will get to eat pizza? Hints: Let A be the 6x6 matrix defined by A=[aij], where aij is the probability that Jack will have $(i1)after one game is he starts with $(j1). For example, a23  .40 since there is a 40% probability that Jack will end up with $1 after a game is he starts the game with $2 (If Jack wins 605 of the time, he must lose 40% of the time). Also, for example, a52 = 0 since there is no way jack can have $4 after one game if he had $1 at the beginning of the game. Since Jack will stop if he goes broke or accumulates $5, a11 and a66 are both 1.
Let x0 = [0 0 1 0 0 0 ] transposed, which we interpret as saying that initially Jack has $2 with a probability 1. Then Ax0 will represent the porbability of each amount of money, $0$5, after one game. What is the probability that Jack will be able to eat pizza by computing Akx0 for large k and finding a limiting value.
Also what is the probability Jack will get to eat pizza if he starts with $3?
Thank you for your help. My daughter knows this is called a monte carlo problem and that limits will have to be used once the 6x6 matrix is set up.
Thank you


Hi Grace,
I set up the matrix according to the instructions and got
A =
1 
0.4 
0 
0 
0 
0 
0 
0 
0.4 
0 
0 
0 
0 
0.6 
0 
0.4 
0 
0 
0 
0 
0.6 
0 
0.4 
0 
0 
0 
0 
0.6 
0 
0 
0 
0 
0 
0 
0.6 
1 
One check is that the sum of the entries in each column must be 1.
To find the probability that Jack will get to eat pizza if he starts with $2 is, as you say in the problem, to find A^{k} for large k and then compute A^{k} x_{0} where x_{0} is the transpose of [0,0,1,0,0,0]. The probability that Jack eats pizza is the entry in the sixth row of A^{k} x_{0} .
I am not sure how your child will find A^{k} .
Penny

