Ask Question
21 April, 03:43

Suppose you wanted to evaluate the performance of the three judges in Smallville, Texas: Judge Adams, Judge Brown, and Judge Carter. Over a three-year period in Smallville, Judge Adams saw 27% of the cases, Judge Brown saw 31% of the cases, and Judge Carter saw the remainder of the cases. 5% of Judge Adams' cases were appealed, 7% of Judge Brown's cases were appealed, and 9% of Judge Carter's cases were appealed. (See the case problem on pages 216-218 of your textbook for a similar problem.) Given a case from this three-year period was not appealed, what is the probability the judge in the case was not Judge Brown? Question 1 Of 46 bank accounts at a small bank, 25 accounts have values of less than $1,000 and the rest have values of at least $1,000. Suppose 4 accounts are randomly sampled (See exercise 9 on page 184 of your textbook for a similar problem.) What is the probability that exactly 3 of the 4 accounts have values of at least $1,000?

+4
Answers (1)
  1. 21 April, 07:35
    0
    Answer 1: Given that the case from this three-year period was not appealed, the probability that the judge was not Judge Brown is 0.689.

    Answer 2: Probability that exactly 3 of the 4 accounts have values of at least $1000 is 0.2068.

    Step-by-step explanation:

    Answer 1:

    Probabilities that each of the judges saw the cases are:

    P (Judge Adams) = 0.27

    P (Judge Brown) = 0.31

    P (Judge Carter) = 1 - (0.27 + 0.31)

    P (Judge Carter) = 0.42

    Probabilities that the cases of each of the judges were appealed (A) can be represented as a conditional probability:

    P (A | Judge Adams) = 0.05

    P (A | Judge Brown) = 0.07

    P (A | Judge Carter) = 0.09

    Hence we can find the probabilities that the cases of each of the judges were not appealed:

    P (Not A | Judge Adams) = 1 - 0.05 = 0.95

    P (Not A | Judge Brown) = 1 - 0.07 = 0.93

    P (Not A | Judge Carter) = 1 - 0.09 = 0.91

    We need to find the probability P (Judge Brown | Not A). For this we will use the Baye's Theorem:

    P (Ai|B) = [P (Ai) P (B|Ai) ]/[P (A1) P (B|A1) + P (A2) P (B|A2) + P (A3) P (B|A3) ]

    P (Judge Brown | Not A) = [P (Judge Brown) * P (Not A | Judge Brown) ]/[P (Judge Adams) * P (Not A | Judge Adams) + P (Judge Brown) * P (Not A | Judge Brown) + P (Judge Carter) * P (Not A | Judge Carter) ]

    P (Judge Brown | Not A) = [0.31*0.93]/[ (0.27*0.95) + (0.31*0.93) + (0.42*0.91) ]

    = 0.2883 / (0.2565 + 0.2883 + 0.3822)

    = 0.2883/0.927

    P (Judge Brown | Not A) = 0.311

    P (Not Judge Brown | Not A) = 1 - 0.311 = 0.689

    Given a case from this three-year period was not appealed, the probability the judge in the case was not Judge Brown is 0.689

    Answer 2:

    We will use the binomial distribution formula to find out the probability that exactly 3 of the 4 accounts have values of at least $1000. The binomial distribution formula is:

    P (X=x) = ⁿCₓ pˣ qⁿ⁻ˣ

    Where n = no. of trials

    x = no. of successful trials

    p = probability of success

    q = probability of failure = 1-p

    We have,

    n = 46

    x = 4

    p = (46-25) / 46 = 21/46

    q = 25/46

    P (X=3) = ⁴C₃ (21/46) ³ (25/46) ⁴⁻³

    = 4 * (21/46) ³ (25/46) ¹

    P (X=3) = 0.2068
Know the Answer?
Not Sure About the Answer?
Find an answer to your question ✅ “Suppose you wanted to evaluate the performance of the three judges in Smallville, Texas: Judge Adams, Judge Brown, and Judge Carter. Over a ...” in 📘 Mathematics if you're in doubt about the correctness of the answers or there's no answer, then try to use the smart search and find answers to the similar questions.
Search for Other Answers