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Problem 7-32 An engineer has designed a modified welding robot. The robot will be considered good enough to manufacture if it misses only 1% of its assigned welds. And it will be judged a poor performer if it misses 5% of its welds. (In-between possibilities are not considered.) A test is performed involving 100 welds. The new design will be accepted if the number of missed welds R is 2 or fewer and rejected otherwise.(a) What is the probability that a good design will be rejected?
Probability=
(b) What is the probability that a poor design will be accepted?
Probability=

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Answer:

a. p= 0.0794 or 7.94%

b. p= 0.1183 or 11.83%

Step-by-step explanation:

a. This is binomial probability function expressed as:

[tex]P(X=x)={n\choose x}p^x(1-p)^{n-x}[/tex]

where:

  • x=is the number of successful events
  • p=probability of success
  • n=the number of events.

The probability of good design being rejected is equal to 1 minus a good design being accepted.

#Given p=0.01, n=100  this probability is calculated as:

[tex]P(X=x)={n\choose x}p^x(1-p)^{n-x}\\\\1-P(X\leq 2)=1-[P(X=2)+P(X=1)+P(X=1)]\\\\=1-[{100\choose 2}0.01^2(0.99)^{98}+{100\choose 1}0.01^1(0.99)^{99}+{100\choose 0}0.01^x(0.99)^{100}]\\\\=1-[0.1849+0.3697+0.3660]\\\\=0.0794[/tex]

Hence, the probability that a good structure is 0.0794 or 7.94%

b. The probability that a bad design is accepted is equivalent to the probability that at least two bad designs are accepted.

Given that p=0.05, n=100 and the x=0,1,2... the probability is:

[tex]P(X\leq 2)=P(X=0)+P(X=1)+P(X=2)\\\\={100\choose 0}0.05^0(0.95)^{100}+{100\choose 1}0.05^1(0.95)^{99}+{100\choose 2}0.05^2(0.95)^{98}\\\\\\=0.0059+0.0312+0.0812\\\\=0.1183[/tex]

Hence, the probability of a bad design being accepted is 0.1183 or 11.83%

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