The prior probabilities for events A₁, A₂, and A₃

are P(A₁) = 0.30,

P(A₂) = 0.20,

and P(A₃) = 0.50.

The conditional probabilities of event B given A₁,

A₂,

and A₃

are P(B | A₁) = 0.40,

P(B | A₂) = 0.30,

and P(B | A₃) = 0.50.

(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(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(A_i)

P(B | A_i)

P(A_i B)

P(A_i | B)

A₁

0.30 0.40 A₂

0.20 0.30 A₃

0.50 0.50

1.00

1.00