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The prior probabilities for eventsA1, A2, and A3areP(A1) = 0.20,P(A2) = 0.30,andP(A3) = 0.50.The conditional probabilities of event B givenA1,A2,andA3areP(B | A1) = 0.50,P(B | A2) = 0.30,andP(B | A3) = 0.40.(Assume thatA1, A2, and A3are mutually exclusive events whose union is the entire sample space.)(a)ComputeP(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 probabilityP(A2 | B).(Round your answer to two decimal places.) (c)Use the tabular approach to applying Bayes' theorem to computeP(A1 | B),P(A2 | B),andP(A3 | B).(Round your answers to two decimal places.)EventsP(Ai)P(B | Ai)P(Ai ∩ B)P(Ai | B)A10.200.50 A20.300.30 A30.500.40 1.00 1.00

Question

Asked Oct 20, 2019

The prior probabilities for events

A₁, A₂, and A₃

are

P(A₁) = 0.20,

P(A₂) = 0.30,

and

P(A₃) = 0.50.

The conditional probabilities of event B given

A₁,

A₂,

and

A₃

are

P(B | A₁) = 0.50,

P(B | A₂) = 0.30,

and

P(B | A₃) = 0.40.

(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)
A₁	0.20	0.50
A₂	0.30	0.30
A₃	0.50	0.40
	1.00		1.00

Expert Answer

Step 1

(a)

From the given information, P(A₁)=0.20, P(A₂)=0.30, P(A₃)=0.50, P(B|A₁)=0.5, P(B|A₂)=0.30 and P(B|A₃)=0.40.

P(Bn4) P(B 4)x P(4)
=0.5x 0.20
0.1
P(Bn4) P(B|A,) x P(A, )
0.30x 0.30
= 0.09
P(Bn4,) P(B|A,)x P(A,)
0.40 x 0.50
0.2 — Image Transcription
P(Bn4) P(B 4)x P(4) =0.5x 0.20 0.1 P(Bn4) P(B|A,) x P(A, ) 0.30x 0.30 = 0.09 P(Bn4,) P(B|A,)x P(A,) 0.40 x 0.50 0.2

Step 2

(b)

The value of P(A2|B) is 0.23 and it is calculated...

P(B A,) P(4)
P(B A) P(A)+P(B| A,) P(A)+ P(B| A,)P(A,)
P(A B)
0.30 x 0.30
+(0.30x 0.30)+(0.40x 0.50)
(0.5x0.20)+
0.09
0.1 0.09 0.2
=0.23 — Image Transcription
P(B A,) P(4) P(B A) P(A)+P(B| A,) P(A)+ P(B| A,)P(A,) P(A B) 0.30 x 0.30 +(0.30x 0.30)+(0.40x 0.50) (0.5x0.20)+ 0.09 0.1 0.09 0.2 =0.23

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