Hi,

yes ur transition matrix is correct.

He will pass the first test,the intial probability distribution is given by A(0)=[1,0].

The probability that regan will pass the fifth test is=A(0)p^4 [i.e if u take transition matix is p then find p^4 matrix]

next u find the matix mltiplication for A(0) and p^4 u get the required result.