AMS 310 hw 2

AMS 310 (Fall, 2023) Prof. Rispoli Homework Set # 2 Due Date & Chapters , September 24, 2023 before midnight. Based on the last part of Chapter 2 and Chapter 3, of Ahn. Typing : You must type up solutions to questions 1 and 2. It is recommended that the other solutions be typed also, but this is not required. Total Points There are 12 problems. All problems are worth 8 points each. Reminder: Show your reasoning! Chapter 2 1. A cell phone company has three different production sites. Five percent of the cars from Site 1, 7% from Site 2, and 9% from Site 3 have been recalled due to unexpected shutdown issue. Suppose that 60% of the phones are produced at Site 1, 30% at Site 2, and 10% at Site 3. If a randomly selected cell phone has been recalled, what is the probability that it came from Site 3 (express the answer up to the second decimal place)? P(getting it from site 3) = P(A ∩ B)/P(B) = ( 0.1 * 0.09) / ( 0.05 * 0.6 + 0.07 * 0.3+0.09 * 1) = 0.15 2. Bowl 1 contains 3 red chips and 7 blue chips. Bowl 2 contains 6 red chips and 4 blue. A single chip is drawn from a randomly selected bowl. a) What is the probability that the chip is red? b) Given that the chip is red, what is the probability it came from bowl 2? c) Let A be the event the chip is red, and let B be the event the chip came from Bowl 2. Are these events independent? Explain your answer. a) 9/20 = 0.45 b) P(A ∩ B)/P(B) = (0.6*0.5)/0.45 = 8/3 = 0.67 c) P(A ∩ B)/P(B) = 0.6 P(A) = 0.45 Event A is not independent of event B, as the occurrence of B affects the probability of A, making ? ( 𝐴∣? ) ≠ ? ( 𝐴 ) 3. Let X be a random variable with cdf

F ( x ) = { 0 x < 0 1 10 0 ≤x < 1 3 10 1 ≤ x < 2 6 10 2 ≤ x < 3 1 x ≥ 3 a) Find the probability distribution of X. b) Find the expected value E(X). c) Find the standard deviation of X. X 0 1 2 3 F(x) 1/10 2/10 3/10 4/10 b) E(X) = (0 * 0.1) + ( 1 * 0.2) + (2 * 0.3) + (3 * 0.4) = 2 c) E(x^2) = 0^2 * 0.1 + 1^2 * 0.2 + 2^2 * 0.3 + 3^2 * 0.4 = 5 var = 5 – 2^2 = 1 standard deviation of X = √ 1 = 1 4. Fifteen percent of an airline’s current customers qualify for an executive traveler’s club membership. a) Find the probability that between 2 and 5 (inclusive) out of 20 randomly selected customers qualify for the membership. 0.7571 b) Find the expected number and the standard deviation of the number who qualify in a randomly selected sample of 60 customers. E(x) = 60 * 0.15 = 9 Vars = 60 * 0.15(1 – 0.15) = 7.65 Standard deviation = √ 7.65 = 2.76

Z = (230 – 180)/20 = 2.5 Lower = -2.5 Upper = 2.5 Mean = 0 Standard Deviation = 1 So, normalcdf = 0.988 8. Use R to answer the following question. Copy and paste the code and answer from R into your paper. In a class of students, 25% have hardcover and 75% students have paperback textbooks. If you randomly choose 50 students in this class with replacement, find the probability that at most 10 of them have hardcover texts. 9. A recent national study showed that approximately 44.7% of college students have used Wikipedia as a source in at least one of their term papers. Let X equal the number of students in a random sample of size n = 25 who have used Wikipedia as a source. Use R to answer parts b, c, d and e. a) What is the distribution for X. Give the distribution name and values for the parameters. b) Find the probability that X is equal to 17. c) Find the probability that X is at most 13. d) Find the probability that X is bigger than 11. e) Find the probability that X is between 16 and 19, inclusive. a) X ~ Bin(25,0.447) n = 25 p = 0.447 b) c) d)

e) 10. Consider the following questions: give the distribution name, carefully define the random variables you use and find the final answer. a) Suppose a representative at a credit card customer service center receives a phone call every 5 minutes on average. Find the probability that she receives 3 phone calls in 20 minutes. b) Products produced by a machine has a 3% defective rate. What is the probability that the first defective occurs in the fifth item inspected. c) In a class of 100 students, 25 have hardcover and 75 students have paperback textbooks. If you randomly choose 10 students in this class, find the probability that 2 of them have hardcover texts. d) Emily hits 60% of her free throws in basketball games. She had 7 free throws in last week’s game. What is the probability that she made at least 5 hits? a) X = no. of phone calls the person receives in 20 mins interval λ = 20/5 = 4 calls x = 3 ( given) P(getting 3 calls in 20 mins) = (4^3 * e^-4)/3! = 0.19536 b) p = 0.03 ( given) n= 5 P(that the first defective item occurs in the fifth item inspected) = {( 1 – 0.03)^(5-1)} * 0.03 = 0.026558 c) p = 0.25 ( given) n =10 ( given) P( getting a student with 2 of them have hardcover texts) = 2 C 10 * 0.12 * 0.75^8 = 0.28 11. Computers from a particular company are found to last on average for three years without any hardware malfunction, with a standard deviation of two months. At least what percent of the computers last between 31 months and 41 months? Standard deviation = 2 Mean = 36