Problems

1. (10 points) Textbook problem 4.2
2. (10 points) Textbook problem 4.6. Express your answers using choose and/or summation notation. There's no need to compute numerical answers.
3. (10 points) Textbook problem 4.7
4. (10 points) A student invests on three risky projects. He will spend $100 on each project. Let S1, S2 and S3 be the events that the first, second and third project respectively return some money. Suppose all events are mutually independent and \( p_1 = P(S1)=0.2 \) and \( S1 \) returns $1000, \( p_2 = P(S2)=0.3 \) and \( S2 \) returns $500, \( p_3 = P(S3)=0.6 \) and \( S3 \) returns $200. Let \(W \) be the random variable that represents the student's net returns in dollars.
   1. Calculate the possible values for \(W \) and the corresponding probabilities
   2. What is the expected value of the winnings, \(E[W] \) ?
5. (10 points) Textbook problem 4.23. Do both parts of this problem. For part (b) of this problem, simulate with \( N = 10^5 \) trials and submit your answer as a graph (plot) of a function of \( p \) where \( p = 0.1, 0.2, \ldots, 0.9, 1 \). Submit your code for this problem by pasting it in your pdf, not in a separate file.
6. (2 extra points) Textbook problem 4.14.