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Probability question

2023 · 8 Apr · Shift 1 · Q26
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  5. /2023 · 8 Apr · Shift 1 · Q26

Probability question

2023 · 8 Apr · Shift 1 · Q26

JEE MainMathematicsProbabilityMCQ+4 / −1
In a bolt factory, machines A,BA, BA,B and CCC manufacture respectively 20%,30%20 \%, 30 \%20%,30% and 50%50 \%50% of the total bolts. Of their output 3, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found the defective, then the probability that it is manufactured by the machine CCC is :
  1. A
    27\frac{2}{7}72​
  2. B
    928\frac{9}{28}289​
  3. C
    514\frac{5}{14}145​
  4. D
    37\frac{3}{7}73​
View written solutionFree

Correct answer: C

  1. Define the events

Let:

  • AAA = bolt is manufactured by machine AAA
  • BBB = bolt is manufactured by machine BBB
  • CCC = bolt is manufactured by machine CCC
  • DDD = bolt is defective

Given: P(A)=0.20,P(B)=0.30,P(C)=0.50P(A)=0.20, \quad P(B)=0.30, \quad P(C)=0.50P(A)=0.20,P(B)=0.30,P(C)=0.50

Defective rates: P(D∣A)=0.03,P(D∣B)=0.04,P(D∣C)=0.02P(D|A)=0.03, \quad P(D|B)=0.04, \quad P(D|C)=0.02P(D∣A)=0.03,P(D∣B)=0.04,P(D∣C)=0.02

We need to find: P(C∣D)P(C|D)P(C∣D)

This is a direct application of Bayes' theorem.

  1. Find total probability of a defective bolt

Using total probability: P(D)=P(A)P(D∣A)+P(B)P(D∣B)+P(C)P(D∣C)P(D)=P(A)P(D|A)+P(B)P(D|B)+P(C)P(D|C)P(D)=P(A)P(D∣A)+P(B)P(D∣B)+P(C)P(D∣C)

Substitute values: P(D)=0.20⋅0.03+0.30⋅0.04+0.50⋅0.02P(D)=0.20\cdot 0.03+0.30\cdot 0.04+0.50\cdot 0.02P(D)=0.20⋅0.03+0.30⋅0.04+0.50⋅0.02

P(D)=0.006+0.012+0.010=0.028P(D)=0.006+0.012+0.010=0.028P(D)=0.006+0.012+0.010=0.028

  1. Apply Bayes' theorem

P(C∣D)=P(C)P(D∣C)P(D)P(C|D)=\frac{P(C)P(D|C)}{P(D)}P(C∣D)=P(D)P(C)P(D∣C)​

P(C∣D)=0.50⋅0.020.028P(C|D)=\frac{0.50\cdot 0.02}{0.028}P(C∣D)=0.0280.50⋅0.02​

P(C∣D)=0.010.028=1028=514P(C|D)=\frac{0.01}{0.028}=\frac{10}{28}=\frac{5}{14}P(C∣D)=0.0280.01​=2810​=145​

  1. Match with the options

514\frac{5}{14}145​ corresponds to Option C.

  1. Verification with stored answer

Stored correct answer: C

Our derived answer: C

So they agree.

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