What is the answer of 23 please explain me why Redo exercise

What is the answer of 2,3? please explain me why

Redo exercises 15 and 16 in section 8.1 of your textbook, about the volume of securities sold by a brokerage firm, using the following data: If volume is high this week, then next week it will be high with a probability of 0.9 and low with a probability of 0.1. If volume is low this week then it will be high next week with a probability of 0.3. Assume that state 1 is high volume and that state 2 is low volume. Find the transition matrix for this Markov process. If the volume this week is high, what is the probability that the volume will be high two weeks from now? What is the probability that volume will be high for three consecutive weeks?

Solution

(2) If volume is high this week then there can be two cases:

(i) First case: Next week is low and second week is high.

Probability in this case = 0.1*0.3 = 0.03

(ii) Second Case: Next week is high and second week is high.

Probability in this case = 0.9*0.9 = 0.81

   Therefore, total probability = 0.03 + 0.81 = 0.84

(3) For volume to be high for 3 consecutive weeks, we assume we have low volume in this week:

Then probability for next week will be = 0.3

Now, probability for volume to be high in the next to next week = 0.3*0.9 = 0.27

Now, probability for volume to be high in the next to next to next week i.e. in 3 consecutive weeks = 0.27*0.9 = 0.243

Hence Proved with full explanation.