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```1 probability
Specify Sample Space
• 1-1: Toss a coin two times and note the sequence
• 1-2: Toss a coin three times and note the number
• 1-3: Pick two numbers at random between zero
and one.
• 1-4: Pick a number X at random between zero
and one, then pick a number Y at random
between zero and X.
HW1-1 (to be posted)
• Three systems are depicted, each consisting of 3
unreliable components. The series system works if and
only if (abbreviated as iff) all components work; the
parallel system works iff at least one of the components
works; and the 2-out-of-3 system works iff at least 2 out
of 3 components work. Find the event that each system is
functioning.
Prove the theorems
• 1-5: P() = 0
• 1-6: A  B => P(A)  P(B)
• 1-7: P(A)  1
• 1-8: P(Ac) = 1 – P(A)
• 1-9: P(AB) = P(A) + P(B) – P(AB)
1-10: Assign the probability
• Probability: The random experiment is to
throw a fair die. How can we define
sample space S, and probability law P to
an arbitrary event E (that belongs to 2S)?
1-11: Find the probability
1-12: Conditional Prob.
• An urn contains two black balls, numbered 1
and 2, and two white balls, numbered 3 and 4.
• S = {(1,b), (2,b), (3,w), (4,w)}
• A: black balls are selected,
• B: even-numbered balls are selected
• C: number of selected ball is greater than 2
• Assuming that the four outcomes are equally
likely, find P[A|B] and P[A|C]
1-13: Bayes’ rule (1/2)
• Young actors are more drug-addictive?
• Sample space is young and old actors
D+
Dyoung
actors
Y
10
20
30
old
actors
O
10
50
60
20
70
90
1-13 Bayes’ rule (2/2)
• P[Y|D+] = P[YD+] / P[D+]
• P[Y] = 30/90
• P[YD+] = P[D+|Y] * P[Y] = 10/30 * 30/90
• P[D+] = P[D+|Y] * P[Y] + P[D+|O] * P[O]
= 10/30 * 30/90 + 10/60 * 60/90
= 20/90
• P[Y] = 30/90
HW 1-2 (to be posted)
• Suppose a drug test is 99% true positive
and 99% true negative results. Suppose
that 0.5% of people are users of the
drug. If a guy tests positive, what is
the probability he is a real drug user?
HW 1-3 (to be posted)
• Go back to young actor drug problem. Which
values are to be in the blanks if we want to
conclude that young actors are not more drugaddictive?
D+
Dyoung
actors
old
actors
Y
10
20
30
O
??
??
60
??
??
90
```
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