Review of Set Operation The mathematical basis of probability is the theory of sets.
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Transcript of Review of Set Operation The mathematical basis of probability is the theory of sets.
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 Algebra Probability
Set Event
Universal set Sample space
Element Outcome
Finest-grain: All possible distinguishable outcomes are identified separately
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 ? (0.3+0.15+0.12)
Ex
工廠有 4 部機器生產同一產品,令其為機器A1,A2,A3,A4 。各機器出產產品數量各佔總產量之比為 0.4 ,0.3 ,0.2 ,0.1。再令產品為不良品的事件為 B。各部機器產品的不良率分別為0.02,0.05,0.01,0.02試問若隨機抽取一產品,其為不良品的機率為?
ans.所欲求之不良品的機率即為 P(B),且依題目所示可知若隨機抽取一產品,則 P(A1)=0.4, P(A2)=0.3, P(A3)=0.2,P(A4)=0.1, P(B|A1)=0.02, P(B|A2)=0.05,P(B|A3)=0.01,P(B|A4)
=0.02
P(B)=
4
1
)(i
iAP P(B|Ai)
= P(A1)P(B|A1) + P(A2)P(B|A2) + P(A3)P(B|A3) +P(A4)P(B|A4)
=0.40.02+0.30.05+0.20.01+0.10.02
=0.027
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
,,, %)(%)(%)( 203545 321 BPBPBP
,,, %)|(%)|(%)|( 524 321 BAPBAPBAP
)|()()|()()|()()( 332211 BAPBPBAPBPBAPBPAP
514.0035.0
44.045.0
)(
)()|( 1
1
AP
ABPABP
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?
0002.020/150000/15.0
)()()|(
)|( SP
MPMSPSMP
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 0.95. 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 0.0004 . 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|BC)=0.1• P(B)=0.0004 • P (B|A) =
)|()()|()(
)|()()|(
BAPBPBAPBP
BAPBPABP
...).(..
..00380
100004019500004095000040
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
,)()|(
)()|(
)(
)()|()|(
1
n
iii
iiiii
APABP
APABP
BP
APABPBAP
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)