The prior probabilities for events

A₁,A₂, andA₃

are

P(A₁) =0.50,

P(A₂) =0.30,

and

P(A₃) =0.20.

The conditional probabilities of eventBgiven

A₁,

A₂,

and

A₃

are

P(B|A₁) =0.30,

P(B|A₂) =0.50,

and

P(B|A₃) =0.20.

(Assume that

A₁,A₂, andA₃

are mutually exclusive events whose union is the entire sample space.)(a)Compute

P(BA₁),P(BA₂), andP(BA₃).

P(BA₁)

=

P(BA₂)

=

P(BA₃)

= (b)Apply Bayes' theorem,

P(A_i|B) =

P(A_i)P(B|A_i)

P(A₁)P(B|A₁) +P(A₂)P(B|A₂) ++P(A_n)P(B|A_n)

,

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(AiB)	P(Ai|B)
A1	0.50	0.30
A2	0.30	0.50
A3	0.20	0.20
	1.00			1.00