[Solved] 2 Probability Theory 18 Points 1 4 Points Suppose Biased E Loaded 6 Sided Die 1 3 Appear T Q37231841
2. Probability Theory (18 points) (1) (4 points) Suppose we have biased (i.e. loaded) 6-sided die where 1 and 3 appear twice as often as any other number. The other numbers are all equally likely (a) What probabilities do we assign to each of the outcomes? (b) What is the probability that an odd number appears when we roll this die? (2) (2 points) What is the probability of these events when we randomly select a permutation of 1, 2,3]? (a) 1 precedes 3 (b) 3 precedes 1 (c) 3 precedes 1 and 3 precedes 2 (3) (2 points) Suppose that E and F are events such that p(E)-0.8 and p(F) = 0.6. Show that p(EUR) > 0.8 and p(En F) 0.1 4) (2 points) What is the conditional probability that exactly four heads appear when a fair coin is flipped five times, given that the first flip came up tails? (5) (2 points) Let E be the event that a randomly generated bit string of length three contains an odd number of 1s, and let F be the event that the string starts with 1. Are E and F independent? (6) (6 points) A group of six people play the game of “odd person out” to a person flips a fair coin. If determine who wil buy refreshments. Each there is a person whose outcome is not the same as that of any other member of the group, this person has to buy the refreshments. What is the probability that there is an odd person out after the coins are flipped once? Show transcribed image text 2. Probability Theory (18 points) (1) (4 points) Suppose we have biased (i.e. loaded) 6-sided die where 1 and 3 appear twice as often as any other number. The other numbers are all equally likely (a) What probabilities do we assign to each of the outcomes? (b) What is the probability that an odd number appears when we roll this die? (2) (2 points) What is the probability of these events when we randomly select a permutation of 1, 2,3]? (a) 1 precedes 3 (b) 3 precedes 1 (c) 3 precedes 1 and 3 precedes 2 (3) (2 points) Suppose that E and F are events such that p(E)-0.8 and p(F) = 0.6. Show that p(EUR) > 0.8 and p(En F) 0.1 4) (2 points) What is the conditional probability that exactly four heads appear when a fair coin is flipped five times, given that the first flip came up tails? (5) (2 points) Let E be the event that a randomly generated bit string of length three contains an odd number of 1s, and let F be the event that the string starts with 1. Are E and F independent? (6) (6 points) A group of six people play the game of “odd person out” to a person flips a fair coin. If determine who wil buy refreshments. Each there is a person whose outcome is not the same as that of any other member of the group, this person has to buy the refreshments. What is the probability that there is an odd person out after the coins are flipped once?
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Answer to 2. Probability Theory (18 points) (1) (4 points) Suppose we have biased (i.e. loaded) 6-sided die where 1 and 3 appear t… . . .
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