It’s the same game as before with the same rules: in each round a fair coin...
It’s the same game as before with the same rules: in each round a fair coin is tossed and if it comes up heads you win $1, and if it comes up tails you lose $1. The game consists of 50 such rounds. Your net gain at the end of the game is defined as the total amount of money won by you during the game minus the total amount of money lost by you during the game. Having studied random variables and expecations in 206 recently, you are eager to apply this new knowledge to make the most of amount of money in the least amount of time, and so you come up with the following strategy: you will only play the game if your expected net gain is positive.
Should you play the game? Why or why not?
Suppose that there is another table with the same game as above except that they use a biased coin there: the coin lands heads with probability 5/8 and tails with probability 3/8. Should you play at this table? Why or why not? Give details for all the steps of your solution.
Homework Answers
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It’s the same game as before with the same rules: in each round
a fair coin...
Problem 2: Tails and (Heads or Tails?) Alice and Bob play a coin-tossing game. A fair...
Problem 2: Tails and (Heads or Tails?) Alice and Bob play a coin-tossing game. A fair coin (that is a coin with equal probability of 1. The coin lands 'tails-tails' (that is, a tails is immediately followed by a tails) for the first 2. The coin lands 'tails-heads (that is, a tails is immediately followed by a heads) for the landing heads and tails) is tossed repeatedly until one of the following happens time. In this case Alice wins. first...
71. A game involves selecting a card from a regular 52-card deck and tossing a coin....
71. A game involves selecting a card from a regular 52-card deck and tossing a coin. The coin is a fair coin and is equally likely to land on heads or tails. • If the card is a face card, and the coin lands on Heads, you win $6 • If the card is a face card, and the coin lands on Tails, you win $2 • If the card is not a face card, you lose $2, no matter...
Consider a game in which a coin will be flipped three times. For each heads you will be paid $100. Assume that the...
Consider a game in which a coin will be flipped three times. For each heads you will be paid $100. Assume that the coin comes up heads with probability 1/3. a. Construct a table of the possibilities and probabilities in this game. Probability Outcome Possibilities 0 heads, 3 tails / 1 heads, 2 tails 2 2 heads, 1 tails 3 3 heads, 0 tails b. Compute the expected value of the game. The expected value of the game is $...
Consider a game in which a coin will be flipped three times. For each heads you...
Consider a game in which a coin will be flipped three times. For each heads you will be paid $100. Assume that the coin comes up heads with probability 1/3. a. Construct a table of the possibilities and probabilities in this game. Probability Outcome Possibilities 0 heads, 3 tails / 1 heads, 2 tails 2 2 heads, 1 tails 3 3 heads, 0 tails b. Compute the expected value of the game. The expected value of the game is $...
Suppose that a fair coin is tossed ten times. Each time it lands heads you win...
Suppose that a fair coin is tossed ten times. Each time it lands heads you win a dollar, and each time it lands tails you lose a dollar. Calculate the probability that your total winnings at the end of this game total two dollars, and the probability that your total winnings total negative two dollars.
Suppose you can place a bet in the following game. You flip a fair coin (50-50...
Suppose you can place a bet in the following game. You flip a fair coin (50-50 chance it lands heads). If it lands heads, you get 4 dollars, if it lands tails, you pay 1 dollar. This is the only bet you can make. If you don't make the bet you will neither gain nor lose money. What is the utility for you of the coin landing tails if you make the bet (assume utility is dollars)?
Suppose you can place a bet in the following game. You flip a fair coin (50-50...
Suppose you can place a bet in the following game. You flip a fair coin (50-50 chance it lands heads). If it lands heads, you get 4 dollars, if it lands tails, you pay 1 dollar. This is the only bet you can make. If you don't make the bet you will neither gain nor lose money. Should you place the bet?
35. You and I play the following game: I toss a coin repeatedly. The coin is...
35. You and I play the following game: I toss a coin repeatedly. The coin is unfair and P(H) = p. The game ends the first time that two consecutive heads (HH) or two consec- utive tails (TT) are observed. I win if (HH) is observed and you win if (TT) is observed. Given that I won the game, find the probability that the first coin toss resulted in heads?
Suppose you have a six sided die. One face is printed with the number 1. Two faces...
Suppose you have a six sided die. One face is printed with the number 1. Two faces are printed with the number 2. Three faces are printed with the number 3. You also have 3 coins: C_1, C_2, and C_3. C_1 will land Heads with probability 1/5. C_2 will land Heads with probability 1/3. C_3 will land Heads with probability 1/2. You roll the die. If the die lands with a 1 face up, flip coin C_1 If the die lands with...
Suppose you have a six sided die. One face is printed with the number 1. Two...
Suppose you have a six sided die. One face is printed with the number 1. Two faces are printed with the number 2. Three faces are printed with the number 3. You also have 3 coins: C_1, C_2, and C_3. C_1 will land Heads with probability 1/3. C_2 will land Heads with probability 1/5. C_3 will land Heads with probability 1/4. You roll the die. If the die lands with a 1 face up, flip coin C_1 If the die...
Problem 2: Tails and (Heads or Tails?) Alice and Bob play a coin-tossing game. A fair coin (that is a coin with equal probability of 1. The coin lands 'tails-tails' (that is, a tails is immediately followed by a tails) for the first 2. The coin lands 'tails-heads (that is, a tails is immediately followed by a heads) for the landing heads and tails) is tossed repeatedly until one of the following happens time. In this case Alice wins. first...
Consider a game in which a coin will be flipped three times. For each heads you will be paid $100. Assume that the coin comes up heads with probability 1/3. a. Construct a table of the possibilities and probabilities in this game. Probability Outcome Possibilities 0 heads, 3 tails / 1 heads, 2 tails 2 2 heads, 1 tails 3 3 heads, 0 tails b. Compute the expected value of the game. The expected value of the game is $...
Consider a game in which a coin will be flipped three times. For each heads you will be paid $100. Assume that the coin comes up heads with probability 1/3. a. Construct a table of the possibilities and probabilities in this game. Probability Outcome Possibilities 0 heads, 3 tails / 1 heads, 2 tails 2 2 heads, 1 tails 3 3 heads, 0 tails b. Compute the expected value of the game. The expected value of the game is $...
Suppose that a fair coin is tossed ten times. Each time it lands heads you win a dollar, and each time it lands tails you lose a dollar. Calculate the probability that your total winnings at the end of this game total two dollars, and the probability that your total winnings total negative two dollars.
35. You and I play the following game: I toss a coin repeatedly. The coin is unfair and P(H) = p. The game ends the first time that two consecutive heads (HH) or two consec- utive tails (TT) are observed. I win if (HH) is observed and you win if (TT) is observed. Given that I won the game, find the probability that the first coin toss resulted in heads?
Suppose you have a six sided die. One face is printed with the number 1. Two faces are printed with the number 2. Three faces are printed with the number 3. You also have 3 coins: C_1, C_2, and C_3. C_1 will land Heads with probability 1/3. C_2 will land Heads with probability 1/5. C_3 will land Heads with probability 1/4. You roll the die. If the die lands with a 1 face up, flip coin C_1 If the die...
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