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An automobile plant contracted to buy shock absorbers from two suppliers $X$ and $Y$ . $X$ supplies $60\%$ and Y supplies $40\%$ of the shock absorbers. All shock absorbers are subjected to a quality test. The ones that pass the quality test are considered reliable. Of $X’s$ shock absorbers, $96\%$ are reliable. Of $Y’s$ shock absorbers, $72\%$ are reliable.

The probability that a randomly chosen shock absorber, which is found to be reliable, is made by $Y$ is

1. $0.288$
2. $0.334$
3. $0.667$
4. $0.720$
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B.

Then by using Bayes' Theorem :
$\text{Probability of Y given R} = \dfrac{\text{Probability of Y and R}}{\text{Probability of R}}$

$= \dfrac{0.4 \times 0.72}{0.4 \times 0.72 + 0.6 \times 0.96}$

$= \dfrac{1}{3} = 0.33$

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@Arjun Sir,

@Bikram Sir,

Why the answer is not a) 0.288

I think this calculation is suitable when we have given in question

The probability that a randomly chosen reliable shock absorber, which is found to be reliable, is made by Y is

but in the question they have asked:-

The probability that a randomly chosen shock absorber, which is found to be reliable, is made by Y is

Means among randomly choosen (whether it could be reliable or unreliable or whether it is of X or of Y).

So the denomenator part should be calculated as:-

0.6*0.96 (from x and reliable)+ 0.6*0.04(x and unreliable) + 0.4*0.72(from y and reliable) + 0.4*0.28(from y and unreliable)

which is equal to 1

and Numerator as 0.4*0.72(from y and reliable)

which is 0.288

what is wrong in this.??

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@shubhanshu

even i got 0.288
+1
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But in the question, it is nowhere written that Probability to select Reliable shock absorber from Y out of All reliable shock absorbers.

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@shubhanshu

The probability that a randomly chosen shock absorber, which is found to be reliable, is made by Y is

i think this statement wants us to find Y out of reliable ones

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Then by using Bayes' Theorem :

I think here we are using conditional probablity only..not bayes' theorem

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The probability that a randomly chosen shock absorber, which is found to be reliable

The chosen shock absorber is already been found to reliable which means we should restrict the sample space to reliable ones only....and from these reliable ones we have to look for Y being reliable!

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@jatin khachane 1 finding reliable one from X and Y is nothing but total probability.

Given that what we found is reliable, what is the probability it is from Y => Baye's theorem. So yes it is based on Baye's theorem.

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