A playoff between team A and team B consists of at most three games. The first teamthat wins two games wins the playoff. Assume that there will be no draw for each game.(a) Draw a tree diagram to display all the ways the playoff can proceed.

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.

Your team is to start the next round of games along with other teams. All the teams will start at Select one: • a. the same point on the Gantt chart b. a randomly system-selected point on the Gantt chart • c. the mid-point of Gantt chart where they left off in previous round and the next milestone point d. the point on the Gantt chart where they left off in previous round

There are 2 teams, each having N players. There will be N rounds played between the 2 teams. In every round, a player from team A plays against a player from team B. The more powerful player wins the game. Given the strength of the players of both teams, you have to find the maximum number of rounds team A can win. Note that a player cannot play more than 1 round.Input FormatThe first line of input contains T - the number of test cases. It's followed by 3T lines. The first line contains the N - the size of the team. The next 2 lines contain N numbers each - the strength of the players of team A and team B respectively.Output FormatFor each test case, print the maximum number of rounds team A can win, separated by a new line.Constraints1 <= T <= 5001 <= N <= 100000 <= A[i], B[i] <= 10000ExampleInput341 5 7 4 3 8 2 10 22 3 10 5 33 7 10 5 20 15 Output201ExplanationTest-Case 1Player with strength 5 in team A can defeat player with strength 3 in team B.Player with strength 7 in team A can defeat player with strength 2 in team B.Test-Case 2No Player in team A can defeat any player in team B.Test-Case 3Player with strength 7 in team A can defeat player with strength 5 in team B.

There are 2 teams, each having N players. There will be N rounds played between the 2 teams. In every round, a player from team A plays against a player from team B. The more powerful player wins the game. Given the strength of the players of both the teams, you have to find the maximum number of rounds team A can win. Note that a player cannot play more than 1 round.Input FormatFirst line of input contains T - number of test cases. Its followed by 3T lines. The first line contains N - size of the team. The next 2 lines contains N numbers each - strength of the players of team A and team B respectively.Constraints1 <= T <= 5001 <= N <= 100000 <= A[i], B[i] <= 10000Output FormatFor each test case, print the maximum number of rounds team A can win, separated by newline.Sample Input 0341 5 7 4 3 8 2 10 22 3 10 5 33 7 10 5 20 15 Sample Output 0201Explanation 0Test Case 1Player with strength 5 in team A can defeat player with strength 3 in team B.Player with strength 7 in team A can defeat player with strength 2 in team B.Test Case 2No Player in team A can defeat any player in team B.Test Case 3Player with strength 7 in team A can defeat player with strength 5 in team B.

In a small football league, teams are awarded 5 points for winning a match, 3 points for a draw and 1 point for losing. The team with the most points after 12 matches wins a prize.Team A win 6 of their matches and have half as many draws. The rest of the matches they lose.Team B win as many matches as Team A lose, but all the rest of their matches are draws and they don’t lose a single game.Team C and D both lose 10 matches, but Team C wins the rest and Team D draws the rest.Team E wins 1 more match than Team A, but draws 2 matches and loses the rest.Which team wins the prize?