Question

A team play a series of games with win, lose, or draw outcomes. The transition probabilities between winning, losing, and drawing are averaged over a long time and treated as independent: Loss Draw Win Loss 0.3 0.4 0.3 Draw 0.4 0.5 0.1 Win 0.2 0.4 0.4 A. If the team wins two games in a row, what is the probability that it will draw its next game? B. On average the team wins 50% of the time, draws 20% of the time, and loses 30% of the time. Given this, what is probability that it will win 3 games from now? C. If the team is coming off a losing streak of 3 games, what is the chance that it will not lose its next 2 games? D. The team travels on two different buses, and the travel times are exponentially-distributed with means µ1 and µ2 hours, respectively. Derive the probability of the first bus arriving first.

Answer #1

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A dodgeball team either wins, ties, or loses each game. If they
lose a game, their probability of winning the next game is 1 and
their probability of losing the next game is 0. If they tie a game,
their probability of winning the next game is 0.8 and their
probability of losing the next game is 0.1. If they win a game,
their probability of winning the next game is 0 and their
probability of losing the next game...

The World Series in baseball is won by the first team to win 4
games in a series of 7. Clearly it cannot go longer than 7 games.
Assume the two teams are evenly matched. (a) What is the
probability that the team to win the first game wins the series?
Explain the reasoning behind your computation in words. (b) Suppose
that a team wins the first two games. What is the chance that it
wins the series?

Two teams A and B play a series of at most five games. The first
team to win these games win the series. Assume that the outcomes of
the games are independent. Let p be the probability for team A to
win each game. Let x be the number of games needed for A to win.
Let the event Ak ={A wins on the kth trial}, k=3,4,5.
(a) What is P(A wins)?
Express the probability with p and k. Show...

Baseball's World Series is a maximum of seven games, with the
winner being the first team to win four games. Assume that the
Atlanta Braves and the Minnesota Twins are playing in the World
Series and that the first two games are to be played in Atlanta,
the next three games at the Twins' ballpark, and the last two
games, if necessary, back in Atlanta. Taking into account the
projected starting pitchers for each game and the home field
advantage,...

5. A professional basketball team, has
won 12 of its last 20 games and it is expected to continue winning
at the same percentage rate. The team’s ticket manager is anxious
to attract a large crowd (filling the team’s basketball arena) to
next week’s game but believes that depends on how well the team
performs tonight against its rival. Based on her past experience,
she assese the probability of drawing a full-arena crowd to be 90
percent should the team...

Baseball's World Series is a maximum of seven games, with the
winner being the first team to win four games. Assume that the
Atlanta Braves and the Minnesota Twins are playing in the World
Series and that the first two games are to be played in Atlanta,
the next three games at the Twins' ballpark, and the last two
games, if necessary, back in Atlanta. Taking into account the
projected starting pitchers for each game and the home field
advantage,...

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