﻿[Script Info] Title: [Events] Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text Dialogue: 0,0:00:00.00,0:00:03.00,Default,,0000,0000,0000,,[Thrun] And again the solution follows directly from the state diagram over here. Dialogue: 0,0:00:03.00,0:00:07.00,Default,,0000,0000,0000,,In the beginning we do know we're in state A Dialogue: 0,0:00:07.00,0:00:09.00,Default,,0000,0000,0000,,and the chance of remaining in A is 0.5. Dialogue: 0,0:00:09.00,0:00:13.00,Default,,0000,0000,0000,,This is the 0.5 over here. We can just read this off. Dialogue: 0,0:00:13.00,0:00:19.00,Default,,0000,0000,0000,,For the next state we find ourselves to be with 0.5 chance to be in A Dialogue: 0,0:00:19.00,0:00:21.00,Default,,0000,0000,0000,,and 0.5 chance to be in B. Dialogue: 0,0:00:21.00,0:00:24.00,Default,,0000,0000,0000,,If we're in B, we transition with certainty to A. Dialogue: 0,0:00:24.00,0:00:26.00,Default,,0000,0000,0000,,That's because of the 0.5. Dialogue: 0,0:00:26.00,0:00:31.00,Default,,0000,0000,0000,,But if we're in A, we stay in A with a 0.5 chance. So you put this together. Dialogue: 0,0:00:31.00,0:00:36.00,Default,,0000,0000,0000,,0.5 probability being in A times 0.5 probability of remaining in A Dialogue: 0,0:00:36.00,0:00:41.00,Default,,0000,0000,0000,,plus 0.5 probability to be in B times 1 probability to transition to A. Dialogue: 0,0:00:41.00,0:00:44.00,Default,,0000,0000,0000,,That gives us 0.75. Dialogue: 0,0:00:44.00,0:00:52.00,Default,,0000,0000,0000,,Following the same logic but now we're in A with 0.75 times a 0.5 probability Dialogue: 0,0:00:52.00,0:00:58.00,Default,,0000,0000,0000,,of staying in A plus 0.25 in B, which is 1 minus 0.75, Dialogue: 0,0:00:58.00,0:01:06.00,Default,,0000,0000,0000,,and the transition's uncertainty back to A as 1, we get 0.625. Dialogue: 0,0:01:06.00,0:01:11.00,Default,,0000,0000,0000,,So now you should be able to take a Markov chain and compute by hand Dialogue: 0,0:01:11.00,0:01:16.00,Default,,0000,0000,0000,,or write a piece of software the probabilities of future states. Dialogue: 0,0:01:16.00,9:59:59.99,Default,,0000,0000,0000,,You will be able to predict something. That's really exciting.