4. Suppose I have 5 different colored pairs of socks. I draw one sock at random...

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4. Suppose I have 5 different colored pairs of socks. I draw one sock at random until I get a matching pair. What is the expected number of socks that I have to draw. (a) Do this by writing down and the probability of each event. (b) Do this using the first fact in the beginning of the test. (c) (Bonus no credit) Do this using the second fact. - Here are some facts that I will cover in class tomorrow. Suppose X is a random variable such that X = {1, 2, 3, 4, 5. ...}. That is, it is a positive integer. Then E[X] = P[X > k]. k=1 Let be the space of observables and P be the measure. The conditional expectation of a random variable X given A is Then the following is true. E[X|A] = X(w)P[{w}|A]. k E[X] = E[X\A]P[Ai]. i=1 4. Suppose I have 5 different colored pairs of socks. I draw one sock at random until I get a matching pair. What is the expected number of socks that I have to draw. (a) Do this by writing down and the probability of each event. (b) Do this using the first fact in the beginning of the test. (c) (Bonus no credit) Do this using the second fact. - Here are some facts that I will cover in class tomorrow. Suppose X is a random variable such that X = {1, 2, 3, 4, 5. ...}. That is, it is a positive integer. Then E[X] = P[X > k]. k=1 Let be the space of observables and P be the measure. The conditional expectation of a random variable X given A is Then the following is true. E[X|A] = X(w)P[{w}|A]. k E[X] = E[X\A]P[Ai]. i=1