Question

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 your steps.

(b) If p=0.5, what is P(A wins)?

Answer #1

Two teams A and B play a series of games until one team wins
four games. We assume that the games are played independently and
that the probability that A wins any game is p and B wins (1-p).
What is the probability that the series ends after...
a) 5 games
b) 6 games
c) 7 games
d) n games

Suppose that two teams are playing a series of games, each team
independently wins each game with 1/2 probability. The final winner
of the series is the first team to win four games. Let X be the
number of games that the two teams have played. Find the
distribution of X.

The National Basketball Association (NBA) championship is a
series of up to 7 games between 2 teams. The first team to win 4
games wins the series. Suppose that the probability that Team A
beats Team B in any given game is p > 0.5, independent of the
outcome of all other games. Let f(n, p) be the probability that
Team A wins the series for a series of n games (n is odd) and a
game-win probability p.
Explain...

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...

In the baseball World Series, two teams play games until one
team has won four games; thus the total length of the series must
be between 4 and 7 games. What is the probability of having a World
Series with a length of 4 games under the assumption that the games
are independent events with each team equally likely to win?

Two teams play a series of games until one of the teams wins n
games. In every game, both teams have equal chances of winning and
there are no draws. Compute the expected number of the games played
when (a) n = 2; (b) n = 3. (To keep track of what you are doing, it
can be easier to use different letters for the probabilities of win
for the two teams).

#5. Suppose that two teams (A and B) play a series of games that
ends when one of them has won 3 games. Suppose that each game
played is, independently, won by team A with probability p. Find
the probability distribution of number of games.

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?

Suppose that two teams (for fun, let’s call them the Domestic
Shorthairs and Cache Cows) play a
series of games to determine a winner. In a best-of-three series,
the games end as soon as one
team has won two games. In a best-of-five series, the games end as
soon as one team has won
three games, and so on. Assume that the Domestic Shorthair’s
probability of winning any one
game is p, where .5 < p < 1. (Notice that...

The baseball World Series is won by the first team to win four
games. Suppose both teams are equally likely to win each game. What
is the probability that the team that wins the first game will win
the series?

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