A,B and C play a game and the chances of their winning it in an attempt are (2/3), (1/2) and (1/4) respectively. A has the first chance, followed by B and then by C. This cycle is repeated till one of them wins the game. Then the probability that B wins is

A bookmaker is attempting to calculate the odds of a football team winning the next three matches (match A, match B and match C). They determine the probability of the team winning each respective match to be                P(A) =0.35  ;    P(B) =  0.65 ;   P(C) = 0.58 Assuming that these three events are independent, determine the probability that the team will(i) win all three matches      (Give your answer correct to 3 decimal places)                                                                                    (ii) win at least one match   (Give your answer correct to 3 decimal places)

In the last round of a chess tournament the final match is between Alice and Diego. The winner is the first player to win three games [sometimes called “best of 5”]. Assume that they are equally matched, so that each player has an equal probability of winning each game. What is the probability that the match will be finished after the first 3 games are played?

Two evenly matched basketball teams (call them A and B) compete in a best-of 7 championships (the first team to win 4 games wins the championship). Once the champion has been determined, no more games are played. In each game, there is a home team and an away team. The home team wins the game with probability p ≥ 1/2, independent of all previous games. Suppose that the first three games will be held at the home of team A and the last 4 (or fewer if they are not needed) are played at the home of team B.(a) Let X be the number of games won by team A out of the first 3 games. Specify the distribution of X.(b) Find the probability that only 4 games are played.(c) Which of the two teams is more likely to win the trophy? Explain why.(d) Give an expression for the probability that team A wins the trophy, and evaluate it when p = 0.55.(e) Let Y be the number of games won by team A. Find the probability mass function for Y.(f) Evaluate the expected number of games won by team A and the expected number of games played when p = 1/2.(g) Observe (via computations or simulation) that when p = 0.55 the expected number of games won by team A is larger than that of team B, even though team B is more likely to win the trophy.

if two players A and B are alternatively throwing a coin and a die together. A player who first throws head and 6 wins the game. If A starts the game, then the probability that B wins the game is

Jack and Mark are about to play a game with the following rules. They take turns rolling a standard fair die, starting with Jack. The game ends when someone rolls a 1 or 2 or 3 or 4. That person wins the game. Find the probability that Jack will win the game