求高手翻译！！不要机器！！紧急！

Question

1. CoinGame #2: You play a new game with coinswhere you flip a coin twice. You win $2for both being “heads”, you win $3 for both being “tails”, and you lose $4otherwise. If you played this game manytimes, what would you expect to be your average amount won or lost pergame?

P(win$2)=1/4
P(win$3)=1/4
P(lose$4)=1/2
Expected value of win: (1/4)(2)+(1/4)(3)=5/4
Expected value of lose: (1/2)(-4)= -2（不知道答对了没，怕题目没理解对）

2. MoreRoulette: You decide that you will spendan entire day playing roulette and just betting on the “1stDozen”. This means that you win the betif any of the numbers 1 through 12 come up, but you lose otherwise. This type of bet has a 2:1 payoff. You decide that you will be making $5 bets.
A) Whatis the expected value of this betting strategy?

B) Ifyou end up making a total of 200 bets of this type, how much money would youexpect yourself to have won or lost at the end of the day?

3. Youplay a game where you flip a coin and roll a die each time you play. If you get a “heads” on the coin, then a rollof 1 or 2 wins $12, a roll of 3, 4, or 5 loses $10, and a roll of 6 winsnothing. If you get a “tails” on thecoin, then a roll of 1, 2, or 6 loses $6 and a roll of 3, 4, or 5 wins $9.
A) Whatis the expected value of playing this game?

B) A“fair” game is a game that has an expected value of 0. How much would I have to charge a player pergame for this to be a “fair” game?

Answer 1

1. 硬币游戏#2：你玩一个得抛两次硬币的新游戏。你赢$2如果两个都是正面，你赢$3如果都是反面，任何其他结果会输$4。如果你玩了这个游戏很多次，你认为你的每场游戏平均输赢是多少？（求期望值）

你的答案缺了一步（或可把最后两步合起来）：5/4-2=-3/4

2. 更多轮盘赌：你决定你会花一整天玩轮盘赌而且只赌’第一打‘。这意思你会赢如果1-12里的任何数字出来，否则会输。这种赌有2:1的回报。你决定你会打$5的赌注。
a)这种赌博策略的期望值是多少？
b)如果你打了200个这种赌注，你会在结尾期待你自己赢或输多少钱？

3. 你玩一个每次得抛一次硬币和掷一个骰子的游戏。如果硬币得到正面，那么掷个1或2会赢$12，掷3,4，或5输$10,然后掷6什么都没有。如果硬币抛到反面，那么掷个1，2或6会输$6，掷3,4，或5赢$9。
a)这种游戏的期望值是多少？
b)一个‘公平’的游戏会有期望值为0。你应该每场游戏收玩家多少费用才能称这是个‘公平’的游戏？