Math 180A - Homework 3 ( Due Friday, Oct 14, 2:00 PM) Readings: Sections 3.1 - 3.4 of the textbook, and topics discussed in lectures 7, 8, and 9 of class. Show FULL JUSTIFICATION for all your answers. Warm up Questions - Do not turn in 1. Problem 2. of Chapter 3. of the textbook. 2. Problem 4. of Chapter 3. of the textbook. 3. Theoretical Exercise 1. of Chapter 3. of the textbook. 4. Theoretical Exercise 2. of Chapter 3. of the textbook. 5. Theoretical Exercise 6. of Chapter 3. of the textbook. Homework Exercises - Turn in 1. For every family with 2 children, the sex of the first and the second child is given by a pair xy (where x and y can be either a boy or a girl), and suppose all 4 possible combinations of xy are equally likely. a. What is the conditional probability that the second child is a girl, given that the first child is a boy. b. The king comes from a family of two children. What is the probability that the other child is his sister? 2. Problem 13. of Chapter 3. of the textbook. (Note that this is yet another way to calculate a probability which we have calculated in class using two different methods.) 3. Suppose three fair coins are flipped, and all possible outcomes are equally probable. Let A be the event that the outcomes of coins 1 and 2 are the same (either both heads or both tails), let B be the event that the outcomes of coins 2 and 3 are the same, and let C be the event that the outcomes of coins 1 and 3 are the same. a. Compute probabilities of A, B, and C. b. Compute probabilities of A B. Are A and B independent? c. Are B and C independent? Are A and C independent? d. Compute the probability of A B C. Are A, B, C independent? 4. Each time Bob competes at a tournament which is held once a year, he wins either a gold medal with probability 0.2, a silver medal with probability 0.5, or a bronze medal with probability 0.3. Suppose he attends this tournament for 10 years, and his performance in different years are independent. What is the probability that he wins a total of 4 gold, 4 silver, and 2 bronze medals. 5. Problem 33. of Chapter 3. 1 6. Problem 37. of Chapter 3. 7. Problem 60. of Chapter 3. 8. (False Positives) A test for a certain rare disease is assumed to be correct 95% of the time. In other words, if a person has the disease, the test results are positive with probability 0.95, and if the person does not have the disease, the test results are negative with probability 0.95. A random person drawn from a certain population has probability 0.001 of having the disease. Given that the person just tested positive, what is the probability that this person has the disease ? Remark: Before you start doing the problem, ask yourself what range you expect the probability to be in and then see if your solution agrees with your intuition. According to the Economist, 80% of those questioned at a leading American hospital substantially missed the correct answer to a question of this type! 9. Suppose that an urn has r red balls and b black balls. A ball is drawn and its color is noted. Then that ball is return to the urn, together with additional c balls of the same color as the drawn ball (for a given c > 0). a. Now another ball is drawn from the urn. Find the probability that this ball is red. b. Suppose this procedure is repeated n times. After that, an (n + 1)th ball is drawn. Find the probability pn+1 that this ball is red. [Hint: follow these steps: i. You have computed p2 in part a. Guess a formula for pn+1 . If necessary, compute p3 as well. ii. Prove your guess by induction: suppose that the procedure is repeated n 1 times. How many balls are in the urn now? Assume that your guess is correct for pn . Now let the procedure be repeated one more time: compute pn+1 and show that your guess is correct for pn+1 . iii. Note that you have proved the induction hypothesis in part a.] c. Do you find your answer surprising or intuitive? (do not turn in) Remark: This has been used to model a variety of phenomena, including the spread of computer viruses and infectious diseases. 10. Fill out the Questionnaire in TritonEd. 2