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Chapter 4 Probability EC255: Managerial Statistics Fall 2018 Tiffany Bayley Learning Objectives •Understand the different ways of assigning probability •Understand and apply marginal, union, joint, and conditional probabilities •Select the appropriate law of probability to solve problems •Solve problems using the laws of probability EC255: Chapter 4 2 Probability Matrices •Clean way to show marginal and intersection probabilities •Easy way to compute union and conditional probabilities EC255: Chapter 4 3 How to Create a Probability Matrix EC255: Chapter 4 4 How to Create a Probability Matrix EC255: Chapter 4 5 How to Create a Probability Matrix EC255: Chapter 4 6 How to Create a Probability Matrix EC255: Chapter 4 7 Example 1: Let’s consider mutual fund managers and their education (MBA from a Top-20 school or not) and how their funds perform (outperform the market or not). Let: •1 denote if the fund manager has an MBA from a Top-20 school and •2 denote if the fund manager does not have an MBA from a Top-20 school. •1 denote if their mutual fund outperforms the market and •2 denote if their mutual fund does not outperform the market. You are given 1 ∩ 1 = 0.11, 2 ∩ 1 = 0.06, 1 ∩ 2 = 0.29 and 2 ∩ EC255: Chapter 4 8 Example 1: Solution a) • 1 denote if the fund manager has an MBA from a Top-20 school and The probability that a fund manager has an MBA from a Top-20 school P(A1) = 0.40 • 2 denote if the fund manager does not have an MBA from a Top-20 school. • 1 denote if their mutual fund outperforms the market and a) • 2 denote if their mutual fund does not outperform the market. P(B2): 0.83 • 1 ∩ 1 = 0.11 • 2 ∩ 1 = 0.06 • 1 ∩ 2 = 0.29 • 2 ∩ 2 = 0.54 EC255: Chapter 4 Total 0.11 0.29 0.40 0.06 0.54 0.60 Total 0.17 0.83 1.00 The probability that a fund will underperform the market. 9 Using the Probability Matrix to Solve Problems… Law of Addition: •Calculate P(Red or Ace) 28/52 Color Type Red Black Total Ace 2 2 4 Non-Ace 24 24 48 Total 26 26 52 EC255: Chapter 4 10 Using the Probability Matrix to Solve Problems… Conditional Probability: •Of the cars on a used car lot • 70% have air conditioning (AC) • 40% have bluetooth (BT) • 20% of the cars have both BT Non-BT Total AC 0.20 0.50 0.70 Non-AC 0.20 0.10 0.30 Total 0.40 0.60 1.00 •What is the probability that a car has bluetooth, given that it has AC ? 0.2/0.7 EC255: Chapter 4 11 Using the Probability Matrix to Solve Problems… Conditional Probability: •Out of a target audience of 2,000,000 • ad A reaches 500,000 viewers • ad B reaches 300,000 viewers • both ads reach 100,000 viewers ad B ad A Non-ad B Total 100,000 500,000 300,000 2,000,000 Non-ad A Total •What is the probability of reaching a person with ad A, given that ad B has been seen already? =(100,000/2,000,000)/(300,000/2,000,000) = 1/3 EC255: Chapter 4 12 Concept Check W2-4b #4 • A survey found 43% of Canadians expect to save more money next year than they did last year • 45% plan to reduce debt next year • Of those who expect to save money, 81% plan to reduce debt next year • A Canadian is selected at random • What is the probability that this person expects to save more money and plans to reduce debt? • P(M) = 0.43 • P(R) = 0.45 • P(R|M) = 0.81 • P(M n R) = P(R|M) x P(M) = 0.81 x 0.43 = 0.3483 a) 0.3483 b) 0.88 c) 0.4345 d) 0.19 EC255: Chapter 4 13 Concept Check W2-4b #5 • A survey found 43% of Canadians expect to save more money next year than they did last year • 45% plan to reduce debt next year • Of those who expect to save money, 81% plan to reduce debt next year • A Canadian is selected at random • What is the probability that this person expects to save more money or plans to reduce debt? a) 0.3483 b) 0.5317 c) 0.5755 d) 0.45 EC255: Chapter 4 • P(M) = 0.43 • P(R)=0.45 • P(R|M) = 0.81 P(M U R) = P(M) + P(R) - P(MnR) = 0.43 + 0.45 – 0.3483 =0.5317 14 • A survey found 43% of Canadians expect to save more money next year than they did last year • 45% plan to reduce debt next year • Of those who expect to save money, 81% plan to reduce debt next year • A Canadian is selected at random • What is the probability that this person neither expects to save more money nor plans to reduce debt? a) 0.3483 b) 0.537 c) 0.5755 d) 0.4683 EC255: Chapter 4 • P(M) = 0.43 • P(R)=0.45 • P(R|M) = 0.81 4 P(M’ n R’) = 1– P(M U R) = 1 – 0.5317 = 0.4683 15