Download Bayes` Theorem SOA Exam P: Bayes sample problems

Bayes’ Theorem
Bayes Theorem is a restatement of the definition of conditional probability combined with the law of total probability.
Conditional Probability: If A, B are events in sample space S then by definition
P (A|B) =
but as P (B|A) =
P (A ∩ B)
P (B)
P (A ∩ B)
⇐⇒ P (A ∩ B) = P (B|A)P (A) this implies from our first equation:
P (A)
P (B|A)P (A)
P (A ∩ B)
=
P (A|B) =
P (B)
P (B)
Now for the Law of Total Probability. If {Ei }, i = 1, ..., n is a partition of S, that is Ei ∩ Ej = ∅, i 6= j and ∪i Ei = S
then for any event B then trivially (B ∩ Ei ) ∩ (B ∩ Ej ) = ∅ i 6= j and we have
X
P (B) = P (B ∩ (∪i Ei )) =
P (B ∩ Ei ).
i
So using this last equation we have Bayes:
P (A|B) =
P (B|A)P (A)
P (B|A)P (A)
P (A ∩ B)
=
=P
P (B)
P (B)
i P (B ∩ Ei )
If A = Ej for some j then we have the usual representation:
P (B|Ej )P (Ej )
P (B|Ej )P (Ej )
P (Ej ∩ B)
=
= P
P (Ej |B) =
P (B)
P (B)
i P (B ∩ Ei )
SOA Exam P: Bayes sample problems
• 19. An auto insurance company insures drivers of all ages. An actuary compiled the following statistics on the company’s
insured drivers:
Age of Driver Prob of accident Portion of company’s clients
16-20
0.06
0.08
21-30
0.03
0.15
31.65
0.02
0.49
66-99
0.04
0.28
A randomly selected driver that the company insures has an accident. Calculate the probability that the driver was
age 16-20. (Ans: 0.16)
• 22. A health study tracked a group of persons for five years. At the beginning of the study 20% were classified as heavy
smokers, 30% as light smokers, and 50% as nonsmokers. Results of the study showed that light smokers were twice as
likely as nonsmokers to die during the five-year study, but only half as likely as heavy smokers.
A randomly selected participant from the study died over the five-year period. Calculate the probability that the
participant was a heavy smoker. (Ans: 0.42)
• 28. A hospital receives 1/5 of its flu vaccine shipments from Company X and the remainder of its shipments from other
companies. Each shipment contains a very large number of vaccine vials.
For Company X’s shipments 10% of the vials are ineffective. For every other company, 2% of the vials are ineffective.
The hopsital tests 30 randomly selected vials from a shipment and finds that one vial is ineffective.
What is the probability that this shipment came from Company X? (Ans: 0.10)