In a game show called “Squad Game”, there are n contestants labelled 1 to n, in a circle.Moving clockwise around the circle, starting with contestant 1, each contestant k who has not yet been eliminated nominates a contestant (which may be themselves) for possible elimination. Contestant k then rolls a fair 6-sided die, and if the result is a 6, then the nominated player is eliminated from the game. Play continues in this way until a fixed number L < n of players have been eliminated.(a) Suppose that we watch this game and observe the labels of the L players eliminated in the order in which they were eliminated. Describe a suitable sample space for this experiment.(b) Suppose instead that we watch such a game and observe only the labels of the L players eliminated. Describe a suitable sample space for this experiment.(c) Suppose instead that we watch such a game and observe only whether player 1 is eliminated. Describe a suitable sample space for this experiment.(d) Suppose that each player nominates a uniformly chosen player among those who have not yet been eliminated (i.e. if j < L players have been eliminated then the player whose turn it currently rolls a fair (n − j)-sided die to determine who they will nominate). Find the probability of each sample point in the three experiments (a)-(c) above.

Recall the game “Squad Game”, Suppose that in each round (independent of the past) contestant 1 nominates a uniformly chosen player other than themselves, while all other players always nominate contestant 1.(a) If L = 1, find the probability that contestant 1 is eliminated.(b) Now suppose that n > L = 3. Find the probability that player 1 is eliminated.

There are n𝑛 coins on the table forming a circle, and each coin is either facing up or facing down. Alice and Bob take turns to play the following game, and Alice goes first.In each operation, the player chooses a facing-up coin, removes the coin, and flips the two coins that are adjacent to it. If there are only two coins left, then one will be removed and the other won't be flipped (as it would be flipped twice). If there is only one coin left, no coins will be flipped. If there are no facing-up coins, the player loses.Decide who will win the game if they both play optimally. It can be proved that the game will end in a finite number of operations, and one of them will win.InputEach test contains multiple test cases. The first line contains the number of test cases t𝑡 (1≤t≤1001≤𝑡≤100). The description of the test cases follows.The first line of each test case contains only one positive integer n𝑛 (1≤n≤1001≤𝑛≤100), representing the number of the coins.A string s𝑠 of length n𝑛 follows on the second line of each test case, containing only "U" and "D", representing that each coin is facing up or facing down.OutputFor each test case, print "YES" if Alice will win the game, and "NO" otherwise.You can output the answer in any case (upper or lower). For example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as positive responses.ExampleinputCopy35UUDUD5UDDUD2UUoutputCopyYESNONO

In a tournament, each of the six teams A,B,C,D,E and F plays one matchagainst every other team. In each round of matches, three take placessimultaneously. A TV station has already decided which match it willbroadcast for each round, as shown in the table. In which round willteam D play against team F?ﻓﻲ ﺑﻄﻮﻟﺔ، ﺑﺤﻴﺚ ﻳﻠﻌﺐ ﻛﻞ ﻓﺮﻳﻖ ﻣﺒﺎﺭﺍﺓ ﻭﺍﺣﺪﺓ ﻓﻘﻂ A,B,C,D,E,F ﺗﺸﺎﺭﻙ ﺳﺖ ﻓﺮﻕﺿﺪ ﻛﻞ ﻓﺮﻳﻖ ﻣﻦ ﺍﻟﻔﺮﻕ ﺍﻷﺧﺮﻯ. ﻓﻲ ﻛﻞ ﺟﻮﻟﺔ ﻣﻦ ﺍﻟﺒﻄﻮﻟﺔ ﻳﺘﻢ ﺇﻗﺎﻣﺔ ﺛﻼﺙ ﻣﺒﺎﺭﻳﺎﺕﻣﺘﺰﺍﻣﻨﺔ )ﻓﻲ ﻧﻔﺲ ﺍﻟﻮﻗﺖ(. ﻗﺮﺭﺕ ﻗﻨﺎﺓ ﺗﻠﻔﺰﻳﻮﻧﻴﺔ ﺑﺚ ﻣﺒﺎﺭﺍﺓ ﻭﺍﺣﺪﺓ ﻣﻦ ﻛﻞ ﺟﻮﻟﺔ ﻛﻤﺎ؟F ﺿﺪ ﺍﻟﻔﺮﻳﻖ D ﻣﻮﺿﺢ ﻓﻲ ﺍﻟﺠﺪﻭﻝ. ﻓﻲ ﺃﻱ ﺟﻮﻟﺔ ﻳﻠﻌﺐ ﺍﻟﻔﺮﻳﻖ2 3 4 515 point problems ﻧﻘﺎﻁ ﻟﻜﻞ ﺳﺆﺍﻝ 51 2 3 4 5A – B C – D A – E E – F A – C

There are 8 players. Each player chooses one of the numbers 2, 3, 4, 5, 6, 7, 8, 9 or 10. The player or players whose number(s) is (are) closest to twice of the average of all numbers chosen wins. The payoff of each player is 1 if she wins (regardless of whether or not there are other winners), and 0 if she loses.What is true in this guessing game? None of the other alternatives is true. For each player, choosing the number 2 is the only strategy that survives iterated elimination of strongly dominated strategies. For each player, choosing the number 10 is the strongly dominant strategy. No player has a strongly dominant strategy. For each player, choosing the number 10 is the only strategy that survives iterated elimination of strongly dominated strategies. Every player has a strongly dominated strategy.

Below is given a board game that Honey and Sunny are playing. They both start from the box showing 'start'. They throw a dice that can show a number from 1 to 6. On the basis of the number that has come on dice, the player who has thrown the dice moves that many steps in the direction as represented by arrows. The boxes on which a player reaches during the game, the numbers on those boxes are added together to get the score of that player. Player reaching box representing 'end' first wins the game. If none of them is able to reach the box representing 'end' in 6 turns, the game results in a draw. Honey has the first turn.If in his first three turns, Honey gets 2, 3 and 6 on the dice, respectively, then what would be his score after three turns?