Playing a game at a carnival, you are asked to choose one of five boxes, which are identical in appearance. Two of the boxes contains a prize and the other three are empty. Once you select a box, the carnie running the game opens one of the other boxes, shows you that it is empty, and gives you the chance to change your mind, i.e. to pick an unopened box other than the one you selected first. Will changing your selection improve your chances of winning or not? Give a complete explanation.

Traditionally, the game of Rock, Paper, Scissors is played with two people. Each of the two people, on the count of 3, reveals their hand in the shape of a rock, a piece of paper, or a pair of scissors as best they can (usually with a closed fist, a flat hand, or their index and middle fingers outstretched to form the V of an open pair of scissors). As the rules go, rock smashes scissors, paper covers rock, and scissors cuts paper. The game can be extended to an arbitrarily large number of, say, nn people, where nn is greater than or equal to 2. If when all nn people reveal their play and only 1 or all 3 options appear, that round is deemed indecisiveindecisive and they just play again. If when all nn people reveal their play precisely two of the three options appear, that round is decisivedecisive, and those with the winning play continue while those with the losing play are eliminated. Play continues until only one person remains.  Assume the game begins with nn players, each of whom…

At a county fair, you pay $2 to play the following game: You are given a fair coin, and two bags standin front of you, labelled Bag A and Bag B. You flip a coin: If your coin lands Heads, you pick a ballfrom Bag A. If your coin lands tails, you pick a ball from Bag B. Each ball has a dollar value written onit, and you win the number of dollars shown on the ball. • Bag A contains four (4) balls in total: Three (3) balls show $1, and one (1) ball shows $5.• Bag B contains three (3) balls in total: Two (2) balls show $2, and one (1) ball shows $3.You WIN money if you leave with more money than you started with. You LOSE money if you leavewith less money than you started with. You BREAK EVEN if you leave with the same money that youstarted with.Let H denote the event that you flip heads, T denote the event that you flip tails, W the event that youwin, and L the event that you lose.(i) You pay to play exactly one game. What is the probability that you WIN given thatyou flipped a Heads?(ii)…

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