Review of Set Operation The mathematical basis of probability is the theory of sets.

Review of Set Operation

De Morgan’s Law

Review of Set Operation

Applying Set Theory to Probability Sample space, Events and Probabilities: Outcome: an outcome of an experiment is any possible observations of that experiment. Sample space: is the finest-grain, mutually exclusive, collectively exhaustive set of all possible outcomes. Event: is a set of outcomes of an experiment. Event Space: is a collectively exhaustive, mutually exclusive set of events. Set AlgebraProbability SetEvent Universal setSample space ElementOutcome Finest-grain: All possible distinguishable outcomes are identified separately

Applying Set Theory to Probability

Example Q: A company has a model of telephone usage. It classifies all calls as L (long), B (brief). It also observes whether calls carry voice(V ), fax (F), or data(D). The sample space has six outcomes The probability can be represented in the table Find the probability of a brief data call(0.08), and the probability of a long call ? ( )

Law of Total Probability

Ex

ans.

ex 設某工廠甲、乙、丙 3 個車間生產同一種產品，產量依次占 全廠的 45%,35%,20% 。且各車間的次品率依次為 4%,2%,5% 。現在從待出廠產品中檢查出 1 個次品，問該產 品是由哪個車間生產的可能性大 ?

Ans Let A denote the event that product is defected. Bi denote the product is product from I-th factory

Ans.

Bayes’ Theorem

36 Example of Bayes Theorem Given: A doctor knows that meningitis causes stiff neck 50% of the time Prior probability of any patient having meningitis is 1/50,000 Prior probability of any patient having stiff neck is 1/20 If a patient has stiff neck, what’s the probability he/she has meningitis?

ex 假定用血清蛋白診斷肝癌, 已知確實患 肝癌者被診斷為有肝癌的概率為 確實不是患肝癌者被診斷為有肝癌的概 率為 0.1. 假設在所有人中患有肝癌的 概率為 現在有一個人被診斷為 患有肝癌，求此人確實為肝癌患者的概 率

ANS A 表示診斷出被檢查者患 有肝癌的事件 B 表示被檢查者確實患有 肝癌的事件。 P(A|B)=0.95 P(A|B C )=0.1 P(B)= P (B|A) =

Ex Let 1-Bi, i = 1, 2, 3, denote the probability that plane will be found upon a search of the i-th region when the plane is in that region. What is the conditional probability that the plane is in the i-th region given that a search of region 1 is unsuccessful?

Ans. Let Ai be the event that the plane is in region i. Let B be the event that a search of region 1 is unsucessful P(A1|B ) =(B1 * 1/3 ) /( B1 * 1/3 + 1*1/3 + 1*1/3) = B1 / (B1 + 2) J = 2, 3 P(Aj |B ) =(1 * 1/3 ) /( B1 * 1/3 + 1*1/3 + 1*1/3) = 1 / (B1 + 2)

Independence

Sequential Experiments and Tree Diagrams