6.5 A high school student has two favorite brands of bag pack labeled X and Y. She continuously chooses between these brands in the following manner. Given that she currently has brand X, the probability that she will buy brand X again is 0.8, and the probability that she will buy brand Y next is 0.2. Similarly, given that she currently has brand Y, the probability that she will buy brand X next is 0.3, and the probability that she will buy brand Y again is 0.7. With respect to the time between purchases, if she currently has a brand X bag pack, the time until the next purchase is exponentially distributed with a mean of 6 months. Similarly, given that she currently has a brand Y bag pack, the time until the next purchase is exponentially distributed with a mean of 8 months.

a. What is the long-run probability that she has a brand Y bag pack?

b. Given that she has just purchased brand X, what is the probability that t months later her last purchase was brand Y?