Both George and Sally buy and sell stocks. Just recently George has invested in one particular stock but won’t tell Sally which stock it is. Sally has narrowed it down to three companies BHQ, Stoneside, and AusBank. From past experience, Sally knows that George will choose BHQ 50% of the time, Stoneside 20% of the time, and AusBank 30% of the time. Historically, Stoneside has made a profit for investors 90% of the time, AusBank 80% of the time, and BHQ 60% of the time.

(a) What is the probability of George making a profit?

(b) Sally finds out that George made a profit. What is the probability that he invested in Stoneside?

Suppose that there are three investment plans available for employees of a company: A, B, and C. An employee can only switch from one plan to another at the end of each year. The probability that someone in Plan A will continue with A is 20%, switch to Plan B is 40%, and switch to Plan C is 40%. The probability that someone in Plan B will switch to plan A is 20% and switch to Plan C is 80%. The probability that someone in Plan C will switch to Plan A is 30%, switch to plan B is 10% and continue with C is 60%.

(a) Write down the transition matrix P .

(b) Let Xndenote the probability vector describing the probabili- ties that an employee uses Plan A, B or C in Year n.

If X0 = [0.5, 0.5, 0], find X1 and X2.