4. (Probability) Answer the following questions (a) I have a fair coin and a two-headed coin. I choose one of the two coins randomly with equal probability and then flip it and record if the outcome is head or tail. What is the sample space, events of interest and probability functions of this experiment? Given that the flip was heads, what is the probability that I flipped the two-headed coin? (b) I am trying to send you a single binary bit, either a 0 or a 1. When I transmit the bit, it goes through a series of 2n relays before it arrives at you. Each relay flips the bit independently with a probability p. What is the probability that you will receive the correct bit? (c) (The German Tank Problem) During WWII, Allied forces used statistics to estimate the production number of German tanks. Here is how the technique works. Assuming the Germans produces N tanks annually, with serial numbers 1, 2, 3, ..., N. Suppose that allied forces captured k tanks with a minimum serial number m and a maximum serial number M. The goal is to come up with a good estimate of N. (See Figure below) Serial No: Samples: (c.1) Now imagine that the samples are evenly spaced throughout the range, with additional samples just outside the range from m to M. What is the average gap A between samples? (c.2) It turns out that a good estimator for Nis M + A. Now assume that the Allied Forces has captured 3 German tanks with serial numbers 39, 128, 194, what is your estimate of the number of German tanks? (Note, this problem has been modified to give you an idea of what you can do with probability. For the rigorous and correct derivation, please google "the German Tank problem".) 4. (Probability) Answer the following questions (a) I have a fair coin and a two-headed coin. I choose one of the two coins randomly with equal probability and then flip it and record if the outcome is head or tail. What is the sample space, events of interest and probability functions of this experiment? Given that the flip was heads, what is the probability that I flipped the two-headed coin? (b) I am trying to send you a single binary bit, either a 0 or a 1. When I transmit the bit, it goes through a series of 2n relays before it arrives at you. Each relay flips the bit independently with a probability p. What is the probability that you will receive the correct bit? (c) (The German Tank Problem) During WWII, Allied forces used statistics to estimate the production number of German tanks. Here is how the technique works. Assuming the Germans produces N tanks annually, with serial numbers 1, 2, 3, ..., N. Suppose that allied forces captured k tanks with a minimum serial number m and a maximum serial number M. The goal is to come up with a good estimate of N. (See Figure below) Serial No: Samples: (c.1) Now imagine that the samples are evenly spaced throughout the range, with additional samples just outside the range from m to M. What is the average gap A between samples? (c.2) It turns out that a good estimator for Nis M + A. Now assume that the Allied Forces has captured 3 German tanks with serial numbers 39, 128, 194, what is your estimate of the number of German tanks? (Note, this problem has been modified to give you an idea of what you can do with probability. For the rigorous and correct derivation, please google "the German Tank problem".)