The prior probabilities for events
A1, A2, and A3
are
P(A1) = 0.50,
P(A2) = 0.20,
and
P(A3) = 0.30.
The conditional probabilities of event B given
A1,
A2,
and
A3
are
P(B | A1) = 0.50,
P(B | A2) = 0.40,
and
P(B | A3) = 0.20.
(Assume that
A1, A2, and A3
are mutually exclusive events whose union is the entire sample space.)
(a)
Compute
P(B ∩ A1), P(B ∩ A2), and P(B ∩ A3).
P(B ∩ A1)
=
P(B ∩ A2)
=
P(B ∩ A3)
=
(b)
Apply Bayes' theorem,
P(Ai | B) =
P(Ai)P(B | Ai)P(A1)P(B | A1) + P(A2)P(B | A2) + + P(An)P(B | An)
,
to compute the posterior probability
P(A2 | B).
(Round your answer to two decimal places.)
(c)
Use the tabular approach to applying Bayes' theorem to compute
P(A1 | B),
P(A2 | B),
and
P(A3 | B).
(Round your answers to two decimal places.)
Events
P(Ai)
P(B | Ai)
P(Ai ∩ B)
P(Ai | B)
A1
0.500.50
A2
0.200.40
A3
0.300.201.00 1.00