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 $(i-1)after one game is he starts with $(j-1).  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

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 x₀ where x₀ 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 x₀ .

I am not sure how your child will find A^k .

Penny

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