Question

Object E is dependent on Objects A and B.

P(A works) = 0.90

P(A fails) = 0.10

P(B works) = 0.90

P(B fails) = 0.10

If Object A works, then Probability of Object E working is 0.6

If Object A fails, then Probability of Object E working is 0.2

If Object B works, then Probability of Object E working is 0.6

If Object B fails, then Probability of Object E working is 0.2

What is the Probability of Object E working?

Answer 1

Solution:-

We have given that,

Define events

A: Object "A" works. Therefore , P(A) = 0.90

A' : Object "A" fails. Therefore,. P(A') = 0.10

B : object"B" works. Therefore, P(B) = 0.90

B' : object "B" fails. Therefore, P(B') = 0.10

E : object "E" works

E' : object "E" fails

And also conditional probability are

If Object A works, then Probability of Object E working is 0.6

That is,. P(E given A) = P(E | A) =0.6

If Object A fails, then Probability of Object E working is 0.2

That is,. P(E given A') = P(E | A') = 0.2

If Object B works, then Probability of Object E working is 0.6

That is,. P(E given B) = P(E | B) = 0.6

And

If Object B fails, then Probability of Object E working is 0.2

That is,. P(E given B') = P(E | B' ) = 0.2

Now,

Question) What is the Probability of Object E working?

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By law of total probability theorems

Therefore,

P(E) = P(E | A)*P(A) + P(E | A')*P(A')

P(E) = 0.6*0.90 + 0.2* 0.10

P(E) = 0.54 + 0.02

OR

By using object B

P(E) = P(E | B)*P(B) + P(E | B')*P(B')

P(E) = 0.6*0.90 + 0.2* 0.10

P(E) = 0.54 + 0.02

Result:-

Probability of Object "E" is working = 0.56