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Conditional Probability

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Question 66

A microchip fabrication facility uses two distinct lithography processes: Machine α \alpha\,α and Machine β\betaβ. After processing, each component is classified as High Performance (HHH), Low Performance (LLL), or Defective (DDD).

The following information describes the distribution of these outcomes:

  • A component is processed by Machine α \alpha\,α with probability k k\,k or Machine β \beta\,β with probability 1−k1-k1−k.
  • For Machine α\alphaα, the probability of a Low Performance outcome is 0.25. No defects are produced by Machine α\alphaα.
  • For Machine β\betaβ, the probability of a High Performance outcome is mmm, and the probability of a Low Performance outcome is 0.15.

Given that the total probability of a component being classified as Low Performance is P(L)=0.21P(L) = 0.21P(L)=0.21:

a.

Find the value of kkk.

[3]
b.

Given also that the conditional probability P(β∣H)=449\displaystyle P(\beta | H) = \frac{4}{49}P(β∣H)=494​:

Find the value of mmm.

[4]
c.

Determine the probability that a randomly selected microchip is defective.

[2]
d.

Given that a specific microchip was found not to be defective:

Find the probability that it was processed by Machine α\alphaα.

[2]

Conditional Probability Questions

  1. A Level
  2. /Maths
  3. /Conditional Probability