Expected value = probability of head * expected win - Probability of tail * Expected loss
= 3/4 * 2 - 1/4 * 1
= $1.25
(a) [15 points] Suppose you have a weighted coin in which heads comes up with probability...
Suppose you have an unfair coin that is weighted so that heads comes up only 30 percent of the time. If you flip the coin 4 times, what is the probability that you obtain at least 3 heads in the 4 flips?
You have a biased coin, where the probability of flipping a heads is 70%. You flip once, and the coin comes up tails. What is the expected number of flips from that point (so counting that as flip #0) until the number of heads flipped in total equals the number of tails?
A coin that comes up heads with probability p is flipped n consecutive times. What is the probability that starting with the first flip there are always more heads than tails that have appeared?
Suppose you flip a fair coin repeatedly until you see a Heads followed by another Heads or a Tails followed by another Tails (i.e. until you see the pattern HH or TT). (a)What is the expected number of flips you need to make? (b)Suppose you repeat the above with a weighted coin that has probability of landing Heads equal to p.Show that the expected number of flips you need is 2+p(1−p)/1−p(1−p)
Problem 6 A box contains 4 coins: • coin 1 has both sides heads. • coin 2 has both sides tails. • coin 3 has both sides tails. • coin 4 is a regular coin (1 side head, other side tails). (a)(3 points) If we randomly choose one coin from the box and flip, what is the probability we get heads? (b)(3 points) If we randomly choose one coin, flip, and it comes up heads, what is the probability it...
You have a biased coin where heads come up with probability 2/3 and tails come up with probability 1/3. 2. Assume that you flip the coin until you get three heads or one tail. (a) Draw the possibility tree. (b) What is the average number of flips? Use the possibility tree, and show your calculation. 2. Assume that you flip the coin until you get three heads or one tail. (a) Draw the possibility tree. (b) What is the average...
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 $...
You have in your pocket two coins, one bent (comes up heads with probability 3/4) and one fair (comes up heads with probability 1/2). Not knowing which is which, you choose one at random and toss it. If it comes up heads you guess that it is the biased coin (reasoning that this is the more likely explanation of the observation), and otherwise you guess it is the fair coin. A) What is the probability that your guess is wrong?
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...