 Quandaries and Queries

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

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 Ak for large k and then compute Ak 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 Ak x0 .

I am not sure how your child will find Ak .

Penny

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