GATE CSE 2012 | Question: 63

Question

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

• A. $0.288$
• B. $0.334$
• C. $0.667$
• D. $0.720$

Comments

Learn Baye's Theorem in 5.05 minutes.

https://www.khanacademy.org/math/ap-statistics/probability-ap/stats-conditional-probability/v/bayes-theorem-visualized

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It wants P( Supply by Y| Reliable)=0.333

P( Reliable | Supply by Y)=0.72

Answer 1

B.

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

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

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

Comments

@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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@ Arjun sir

Why is the answer  not . Option C ( .667)

 According to Bayes' theorm first select which observer X and Y .. so probability  ( .5 )   then apply condition

 Plz tell me sir

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Bayes theorem avoided ++

Answer 2

Given information:

• P(X) = 0.60 (60% from supplier X)
• P(Y) = 0.40 (40% from supplier Y)
• P(Reliable|X) = 0.96 (96% of X's are reliable)
• P(Reliable|Y) = 0.72 (72% of Y's are reliable)

Find: P(Y|Reliable)

Step 1: Find the total probability of getting a reliable shock absorber

Using the Law of Total Probability:

P(Reliable):
= P(Reliable∣X)⋅P(X) + P(Reliable∣Y)⋅P(Y)
= 0.96*0.60 + 0.72*0.40
= 0.576+0.288
= 0.864

Step 2: Apply Bayes' Theorem

P(Y∣Reliable):
= P(Y∩Reliable) / P(Reliable) 
= P(Reliable∣Y)⋅P(Y)​ / P(Reliable) 
= 0.72*0.40 / 0.864
= 0.288 / 0.864
= 1/3
= 0.3333 or 0.334