MBA3 Probability distributions and information

1. MBA3 Probability distributions and information Fred Wenstøp

2. Discrete probability distributions • A series of probabilities pi for all possible states of nature Spi =1 • A probability distribution can be • Theoretical, based on simple but fundamental assumptions • Binomial • Pascal • Poisson • Empirical, based on past experience • Subjective, based on beliefs Fred Wenstøp: MBA3

3. The binomial distribution • How many times will I succeed? • A binomial process: • A series of n independent trials where the outcome each time is either success or failure and a constant probability p for success • The probability of exactly a successes in a binomial process • Excel: • BINOMDIST(a;n;p;0) Fred Wenstøp: MBA3

4. The Pascal distribution • When will it be my turn? • The probability that it will take n trials to get the first success in a binomial process with probability p. • Example: • How many tosses to get on the board i Ludo? • p = 1/6 • See graph: Fred Wenstøp: MBA3

5. The Poisson distribution • How often will disasters happen? • The probability of x occurrences of an event in a certain period when the propensity for the event to occur is constant and equal to l per period. • =POISSON(x;l;0) • Example • Norway has about two oil spills per year in coastal waters. The probability that we will have x spills in a certain year is • =POISSON(x;2;0) Fred Wenstøp: MBA3

6. Empirical or subjective distributions • Based on experience or merely assumed • Example: • Future sales of a consumer good Fred Wenstøp: MBA3

7. Continuous probability distributions:Probability densities • In many situations, any value among an infinite number can in principle occur • In practice, the number depends on how precisely we measure the differences between them • Future stock price • Future employment rates • Future sales • The probability of a particular value is therefore zero • Instead, we use probability densities, where areas are probabilities • Example: The normal distribution Fred Wenstøp: MBA3

8. Cumulative distributions • Probability densities can be represented as cumulative distributions which make them easier to handle • The probability of at least x • y = F(x) • F(a) = P(x<a) • Important parameters • Median m, F(m) = 0.5 • Fractiles • The median is the 50% fractile Fred Wenstøp: MBA3

9. Conditional probabilityThe value of tests • A production process produces defect units with probability 0.1 • If an OK unit is shipped, the reward is 100 • If it is defect, a loss of 160 is incurred • A unit can be reworked at an expense of 40 and becomes OK regardless of previous state • A test with sensitivity 0.7 and specificity 0.8 may be performed before any decision is made • What should you be willing to pay for the test? Production P(OK) = 0.9 P(D) = 0.1 Test P(TD|D) = 0.7 P(TOK|D) = 0.3 P(TOK|OK) = 0.8 P(TD|OK) = 0.2 Fred Wenstøp: MBA3

10. Probability tree to represent conditional probabilities 0.72 Production P(OK) = 0.9 P(D) = 0.1 Test P(TD|D) = 0.7 P(TOK|D) = 0.3 P(TOK|OK) = 0.8 P(TD|OK) = 0.2 TOK 0.8 TD OK 0.18 0.2 0.9 0.03 D TOK 0.1 0.3 TD 0.7 0.07 Fred Wenstøp: MBA3

11. 0.72 0.72 TOK OK 0.8 0.96 TD OK D TOK 0.18 0.9 0.2 0.03 0.04 0.03 0.18 D TOK TD OK 0.1 0.72 0.3 TD 0.75 D 0.7 0.28 0.07 0.07 0.25 Transforming a probability tree to a decision oriented tree Fred Wenstøp: MBA3

12. Decision analysis Rework 60 60 D: 0.1 -160 Ship D: 0.04 OK: 0.9 -160 82.2 100 74 OK: 0.96 89.6 100 Ship Rework 60 82.2 TOK: 0.75 89.6 D: 0.28 -160 Test OK: 0.72 Ship 27.2 100 TD: 0.25 Rework 60 60 Fred Wenstøp: MBA3