Please help to solve the question using formulas without using r, thank you! 8.

Question

Please help to solve the question using formulas without using r, thank you!

8. (20pts) Consider the following questions: give the distribution name, carefully define the random variables you use and find the final answer. (a) (4pts) A professor has made 20 exams of which 8 are hard, 7 are reasonable, and 5 are easy. The exams are mixed up and the professor selects 4 of them at random to give to four sections of the course she is teaching. What is the probability that no sections receive a hard test? (b) (4pts) 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. (c) (4pts) Products produced by a machine has a 3% defective rate. what is the probability that the first defective occurs in the fifth item inspected. (d) (4pts) 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. (e) (4pts) 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?

Explanation / Answer

a)Binomial Distribution
Random variable x is the number of hard tests selected

4 non hard test can be selected from 12 tests, this can be done in 12C4 ways

Required probability = 12C4/20C4 = 0.1022

b) Poisson's Distribution
Random variable x is the number of phone calls received

lambda = 1/5 * 20 = 4

P(X = 3) = e^(-4)*4^3/3! = 0.1954

c) Negative Binomial Distribution
Random variable x is the first occurrance of success

P(X = 5) = (1-0.03)^4 * (0.03) = 0.0266

d) Binomial Distribution
Random variable x is the number of students with hardcover text

p = 25/100 = 0.25

P(X = 2) = 10C2 * 0.25^2 * 0.75^8 = 0.2816

e) Binomial Distribution
Random variable x is the number of hits

p = 0.6

P(X >= 5) = 7C5 * 0.6^5 * 0.4^2 + 7C6 * 0.6^6 * 0.4^1 + 7C7 * 0.6^7 * 0.4^0 = 0.4199

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