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?

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)

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

In a tournament, a team has played 40 matches so far and won 30%30% of them. If they win 60%60% of the remaining matches, their overall win percentage will be 50%50%. Suppose they win 90%90% of the remaining matches, then the total number of matches won by the team in the tournament will be.

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.

Four matches are to be played in a chess tournament. In how many ways can result be decided?