Description

Major: Mechanical Engineering
Patrick Gardocki
1. Die Cast Defect Detection (Ch.2)

Figure 1: Examples of Porosity
Table 1: Classified Parts
Porosity No Porosity Total
Defective 865 748 1613
Not Defective 112 8265 8387
Total 987 9013 10000
Let A denote a part with no porosity and B denote a part being defective.
a) A standard defect rate for a die cast part is less than 10%. Determine if this part meets that standard and what is the defect rate for a part that has no porosity.
Find P(B) and P(A ∩ B)
P the part does not meet the standard defect rate of less than 10%
P If a part has no porosity, its defect rate is 7.48%
b) Find the probability of a part being defective given it has porosity
P(B|A′) = (A′∩B)
c) How many defective parts did not have porosity?
P
d) Find the probability of a part having porosity given it being not defective?
P(A′|B′) = P (PB(′B∩′A)′) = 8387112 = 0.013354
2. ChatGPT Exam Solutions (Ch.3)
ChatGPT has exploded in popularity recently, and people are eager to utilize its potential. Professors at forprofit universities have taken to ChatGPT to create exam questions and solutions to save time and effort. The professors are cheap and only use the free version of ChatGPT which is known to only have a 18% success rate of creating correct solution sets for questions it generates.
a) What is the probability that exactly 9 out of 10 solutions are correct if Chat-GPT is used?
P(X = 9) = 109 (0.82)9(0.18)10−9 = 1.626 × 10−6
dbinom(9,10,.18)
## [1] 1.626546e-06
b) What is the probability that at least 5 out of 10 solutions are correct ifChatGPT is used?
10
P x x=5
pbinom(5,10,.18, lower.tail=FALSE)
## [1] 0.003669387
c) What is the probability of at least 1 student catching the professors errorwithin 1 day if at random, the solution manual is checked twice per day?
λT = 2 P
1-ppois(1,2)
## [1] 0.5939942
λT = 2 × 3 = 6
P
1-ppois(6,6)
## [1] 0.3936972
3. Die Casting Defects – Weibull (Ch.4)
PDF: f(x) = 26..5343 x−6.543.111.53 exp− x−6.543.112.53 CDF: F(x) = 1 − exp−6 (.x43) 2.53
a) Determine the probability of failure before an elongation of 7.2%?
P
pweibull(7.2,2.53,6.43)
## [1] 0.7358676
b) Determine what elongation % A319 can survive 85% of the time?
P(X > x
Solve for x, x = 3.13556 Checking solution:

Figure 2: Weibull probability plots for three aluminum castings
1-pweibull(3.13556,2.53,6.43)
## [1] 0.8500002
c) If 100 castings are tested, what is the probability that 85 will not fail till 2.0%elongation?
Let X be the number of castings that pass at 2% elongation. Then X is a binomial random variable.
P
0.726001
sum(dbinom(85:100,100,1-0.1352596))
## [1] 0.726001
d) What is the mean elongation and variance at which failure occurs?
E
V
6.43*gamma(1+ (1/2.53))
## [1] 5.706834
6.43ˆ2*gamma(1+(2/2.53))-6.43ˆ2*(gamma(1+(1/2.53)))ˆ2
## [1] 5.837331
4. Die Casting Defects (Ch.5)

Figure 3: Various Types of Die Casting Defects
Table 2: Joint Probability Distributions
Humidity Porosity (1) Non-Fill (2) Dimensional (3) Solder (4) OK (5)
Low 0.012 0.08 0.00 0.01 0.108
Medium 0.060 0.09 0.02 0.03 0.100
High 0.170 0.21 0.01 0.09 0.010
a) Determine the margnial probability distributions for each defect type andhumidity level.
Table 3: Marginal Probability Distributions
Humidity Porosity (1) Non-Fill (2) Dimensional (3) Solder (4) OK (5) fh(h)
Low 0.012 0.08 0.00 0.01 0.108 0.21
Medium 0.060 0.09 0.02 0.03 0.100 0.30
High 0.170 0.21 0.01 0.09 0.010 0.49
fd(d) 0.242 0.38 0.03 0.13 0.218 1.00
b) Given it is a high humidity day, what is the probability that the part willhave porosity? Any defect?
P(D = 1|H = 3) = = 0.3469
P(0 < D < 4|H = 3) = 0 . = 0.97959
c) Assume the company decided to run production on only low humidity days,what is the probability that the parts will have no defects?
P(D = 5|H = 1) = = 0.5142
d) Are defects and humidity independent?
∵ f1|3(1) ̸= fd(1) therefore NOT independent
References
[1] https://www.sunrise-casting.com/capability/aluminum-die-casting-production.html
[2] https://link.springer.com/chapter/10.1007/978-3-319-48130-2_26
[3] Gardocki, Patrick. Trial Data for PN 198006. Unpublished
[4] https://www.rapiddirect.com/blog/17-types-of-casting-defects/?rdtm_from=sns