Alice, Bob, and Confucius are bored during recess, so they decide to play a new game. Each of them puts a dollar in the pot, and each tosses a coin. Alice wins if the coins land all heads or all tails. Bob wins if two heads and one tail land, and Confucius wins if one head and two tails land. The coins are fair, and the winner receives a net payment of $2 ($3 - $1 = $2), and the losers lose their $1.What is Confucius' expected payoff? Round your answer to three decimal places

Alice, Bob, and Confucius are bored during recess, so they decide to play a new game. Each of them puts a dollar in the pot, and each tosses a coin. Alice wins if the coins land all heads or all tails. Bob wins if two heads and one tail land, and Confucius wins if one head and two tails land. The coins are fair, and the winner receives a net payment of $2 ($3 - $1 = $2), and the losers lose their $1.

In American roulette, the wheel has 38 number: 00, 0, 1, 2, 3, ..., 35, 36 marked on equally spaced slots. If a player bets $10 on a single number and wins, then the player gains $350 otherwise the player loses the $10.This means, that X is either $350 or -$10.What is the expected value of this game?Group of answer choices$0.53$18.95-$18.95-$0.53

Jose has a deck of 10 cards numbered 1 through 10. He is playing a game of chance.This game is this: Jose chooses one card from the deck at random. He wins an amount of money equal to the value of the card if an even numbered card is drawn. He loses $6 if an odd numbered card is drawn.(a) Find the expected value of playing the game.dollars(b) What can Jose expect in the long run, after playing the game many times?(He replaces the card in the deck each time.)Jose can expect to gain money.Hecanexpecttowindollarsperdraw.Jose can expect to lose money.Hecanexpecttolosedollarsperdraw.Jose can expect to break even (neither gain nor lose money).

Three balanced coins are flipped independently. One of the variables of interest is X, thenumber of heads. Let Y denote the amount of money won on a side bet in the followingmanner. If the first head occurs on the first flip, you win $1. If the first head occurs onthe second flip you win $2 and if the first head occurs on the third flip you win $3. Ifno heads appear, you lose $1 (that is, you win −$1).(a) [3 marks] In a table, list all possible outcomes of the experiment, along with thevalues of X and Y associated with each outcome.(b) [3 marks] Determine the bivariate distribution (that is, the joint probability dis-tribution) of X and Y . You can list the probabilities in a table.(c) [3 marks] Find the probability that fewer than three heads will occur and you willwin $1 or less.(d) [2 marks] Are X and Y independent? Why or why no

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