The probability that an integrated circuit chip will have defective etching (E) is 0.13, the probability that it will have a crack defect (C) is 0.29, and the probability that it has at least one defect is 0.32. a) What is the probability that a newly manufactured chip will have both defects? b) What is the probability that a newly manufactured chip will have either an etching or a crack defect (i.e., only one of the two defects)? c) What is the probability that a newly manufactured chip will have neither etching nor crack defect (i.e., both defects are absent) d) It is known that a newly manufactured chip has crack defect. What is the probability that the chip also has defective etching?

Question

The probability that an integrated circuit chip will have defective etching (E) is 0.13, the probability that it will have a crack defect (C) is 0.29, and the probability that it has at least one defect is 0.32.  a) What is the probability that a newly manufactured chip will have both defects?  b) What is the probability that a newly manufactured chip will have either an etching or a crack defect (i.e., only one of the two defects)?  c) What is the probability that a newly manufactured chip will have neither etching nor crack defect (i.e., both defects are absent)  d) It is known that a newly manufactured chip has crack defect. What is the probability that the chip also has defective etching?

cheekabooy · Accepted Answer

At the given condition:

karton · Accepted Answer

Answer:0.345 or 34.5%Explanation:a) Let&#039;s denote the event of a chip having defective etching as E and the event of a chip having a crack defect as C. We are given the following probabilities:P(E)=0.13 (probability of etching defect)P(C)=0.29 (probability of crack defect)P(E∪C)=0.32 (probability of at least one defect)We want to find the probability that a chip will have both defects, i.e., P(E∩C).We can use the inclusion-exclusion principle to find this probability:P(E∪C)=P(E)+P(C)−P(E∩C)Rearranging the equation, we can solve for P(E∩C):P(E∩C)=P(E)+P(C)−P(E∪C)Substituting the given probabilities into the equation:P(E∩C)=0.13+0.29−0.32Simplifying, we find that the probability that a newly manufactured chip will have both defects is 0.1 or 10%.b) We want to find the probability that a newly manufactured chip will have either an etching or a crack defect, but not both.Using the probability of at least one defect, we can calculate this probability as follows:P(E∪C)=P(E)+P(C)−P(E∩C)Since we want only one of the two defects, we subtract the probability of both defects occurring from the probability of at least one defect:P(E∪C∩¬(E∩C))=P(E∪C)−P(E∩C)Substituting the given probabilities into the equation:P(E∪C∩¬(E∩C))=0.32−0.1Simplifying, we find that the probability that a newly manufactured chip will have either an etching or a crack defect (but not both) is 0.22 or 22%.c) We want to find the probability that a newly manufactured chip will have neither an etching nor a crack defect, i.e., both defects are absent.The probability of neither etching nor crack defect can be calculated as:P(¬E∩¬C)=1−P(E∪C)Substituting the given probability into the equation:P(¬E∩¬C)=1−0.32Simplifying, we find that the probability that a newly manufactured chip will have neither etching nor crack defect is 0.68 or 68%.d) We are given that a newly manufactured chip has a crack defect, i.e., C has occurred. We want to find the probability that the chip also has defective etching, i.e., E has occurred given C.We can use the conditional probability formula:P(E|C)=P(E∩C)P(C)Substituting the given probability into the equation:P(E|C)=0.10.29Simplifying, we find that the probability that a newly manufactured chip, which has a crack defect, also has defective etching is approximately 0.345 or 34.5%.

star233 · Accepted Answer

Step 1:a) Let&#039;s denote the event that a chip has defective etching as E and the event that a chip has a crack defect as C.The probability that a chip has both defects can be calculated using the formula for the intersection of two events:P(E∩C)=P(E)+P(C)−P(E∪C)Given that the probability of defective etching (E) is 0.13, the probability of a crack defect (C) is 0.29, and the probability of at least one defect (E∪C) is 0.32, we can substitute these values into the formula:P(E∩C)=0.13+0.29−0.32Calculating the right side of the equation:P(E∩C)=0.42−0.32=0.10Therefore, the probability that a newly manufactured chip will have both defects is 0.10.Step 2:b) To find the probability that a newly manufactured chip will have either an etching or a crack defect (i.e., only one of the two defects), we need to subtract the probability of having both defects from the probability of having at least one defect.P((E∩¬C)∪(¬E∩C))=P(E∪C)−P(E∩C)Substituting the given probabilities:P((E∩¬C)∪(¬E∩C))=0.32−0.10Calculating the right side of the equation:P((E∩¬C)∪(¬E∩C))=0.22Therefore, the probability that a newly manufactured chip will have either an etching or a crack defect (i.e., only one of the two defects) is 0.22.Step 3:c) The probability that a newly manufactured chip will have neither an etching nor a crack defect (i.e., both defects are absent) can be calculated by subtracting the probability of having at least one defect from 1 (since the complement of having at least one defect is having neither defect).P(¬E∩¬C)=1−P(E∪C)Substituting the given probability:P(¬E∩¬C)=1−0.32Calculating the right side of the equation:P(¬E∩¬C)=0.68Therefore, the probability that a newly manufactured chip will have neither an etching nor a crack defect (i.e., both defects are absent) is 0.68.Step 4:d) If it is known that a newly manufactured chip has a crack defect (C), we can use conditional probability to find the probability that the chip also has defective etching (E).The conditional probability can be calculated using the formula:P(E|C)=P(E∩C)P(C)Substituting the given probabilities:P(E|C)=0.100.29Calculating the right side of the equation:P(E|C)≈0.345Therefore, the probability that a newly manufactured chip has defective etching given that it has a crack defect is approximately 0.345.

alenahelenash · Accepted Answer

a) To find the probability that a newly manufactured chip will have both defects, we can use the formula for the intersection of two events:P(E∩C)=P(E)·P(C)Substituting the given probabilities, we have:P(E∩C)=0.13·0.29=0.0377Therefore, the probability that a newly manufactured chip will have both defects is 0.0377.b) To find the probability that a newly manufactured chip will have either an etching or a crack defect (i.e., only one of the two defects), we can use the formula for the union of two mutually exclusive events:P(E∪C)=P(E)+P(C)−P(E∩C)Substituting the given probabilities, we have:P(E∪C)=0.13+0.29−0.0377=0.3823Therefore, the probability that a newly manufactured chip will have either an etching or a crack defect (i.e., only one of the two defects) is 0.3823.c) To find the probability that a newly manufactured chip will have neither an etching nor a crack defect (i.e., both defects are absent), we can subtract the probability of having at least one defect from 1:P({neither&amp;nbsp;E&amp;nbsp;nor&amp;nbsp;C})=1−P(E∪C)Substituting the previously calculated probability, we have:P({neither&amp;nbsp;E&amp;nbsp;nor&amp;nbsp;C})=1−0.3823=0.6177Therefore, the probability that a newly manufactured chip will have neither an etching nor a crack defect (i.e., both defects are absent) is 0.6177.d) To find the probability that a newly manufactured chip has defective etching given that it has a crack defect, we can use the formula for conditional probability:P(E|C)=P(E∩C)P(C)Substituting the given probabilities, we have:P(E|C)=0.03770.29≈0.1303Therefore, the probability that a newly manufactured chip has defective etching given that it has a crack defect is approximately 0.1303.

The probability that an integrated circuit chip will have defective etching (E) is 0.13, the probability that it will have a crack defect (C) is 0.29

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