The team's performance in any given game is correlated to its morale. If the team has won the past two games, then it has a 0.7 probability of winning the next game. If it lost the last game but won the one before that, it has a 0.4 probability of winning. If it won its last game but lost the one before that it has a 0.6 probability of winning. Finally if it lost the last two games it has only a 0.3 probability of winning the next game. No game can end up in a draw. Consider a starting time when the team has won its preceding two games.
1.Show the transition probabilities 2. Find the probability that the first future loss will be followed by another loss. 3. Let X be the number of games played up to, but not including the first loss. Find the probability mass function of X 4. Why do the steady-state probabilities exist? 5. Determine the steady-state probabilities. 6. Find a good approximation to the probability that the team will win its 1000th game, given that the outcomes of games 1000 and 1001 are the same. Clearly state any assumptions you use in the approximation.