Contamination problems in semiconductor manufacturing can...

Contamination problems in semiconductor manufacturing can result in a functional defect, a minor defect, or no defect in the final product. Suppose that $20,50,$ and $30 \%$ of the contamination problems result in functional, minor, and no defects, respectively. Assume that the defects of 10 contamination problems are independent. (a) What is the probability that the 10 contamination problems result in two functional defects and five minor defects? (b) What is the distribution of the number of contamination problems that result in no defects? (c) What is the expected number of contamination problems that result in no defects?

Contamination problems in semiconductor manufacturing can result in a functional defect, a minor defect, or no defect in the final product. Suppose that $20,50,$ and $30 \%$ of the contamination problems result in functional, minor, and no defects, respectively. Assume that the defects of 10 contamination problems are independent.
(a) What is the probability that the 10 contamination problems result in two functional defects and five minor defects?
(b) What is the distribution of the number of contamination problems that result in no defects?
(c) What is the expected number of contamination problems that result in no defects?

Key Concepts

Multinomial Distribution

The multinomial distribution is a generalization of the binomial distribution for experiments where each independent trial can result in more than two outcomes. It calculates the probability of a particular combination of counts for each outcome over a fixed number of trials, using a multinomial coefficient to account for the different arrangements of outcomes.

Binomial Distribution

The binomial distribution describes the number of successes in a fixed number of independent trials when each trial has exactly two possible outcomes (success and failure). It is characterized by a fixed number of trials and a constant probability of success on each trial, and it also provides formulas for calculating probabilities and the expected number of successes.

Expectation in Probability

The expectation of a random variable is the long-run average value that is to be expected if an experiment is repeated many times. It is computed as the sum of each possible value of the random variable multiplied by its probability, and it provides a measure of the central tendency of the distribution.

Transcript

00:01 Hi guys, from the information we can observe that combination problems in semiconductor manufacturing can result a functional defect, a minor defect, or not affect in the final product.

00:13 So suppose there are 20%, 50 % and 30 % of the combination problems results in the functional minor and not effects respective.

00:25 So here we have 24 functional, okay.

00:31 20%, so this is percent, then 50 % for the minor, and 30 % for no defects.

00:46 Okay, so assume that the defects of 10 contamination problems are independent, so let x be the random variable that represents the number of contamination problems that result in the functional, okay, and y, be the random variable that represents the number of contamination problems that results in the minor, and z by the same way for no effects.

01:17 Okay, so the random variables x, y, and z have a multinomial distribution with parameters n equal 10 p1 equal 0 .2 p2 equal 0 .5 and p3 equals 0 .3.

01:39 So the joint probability mass function we can calculate it such as probability of x equals x, y equals y and z equals z so it's n factorial over x factorial y factorial x factorial x x x x x x times p1 power x times p2 power y times p2 power y times p 3 power z okay so this is 10 factorial over x factorial y factorial z fictorian x 0 .2 power x times 0 .5 power y times 0 .3 power z okay so now calculate the probability that the 10 contamination of problems in part a we need to calculate the 10 contamination problems results in two functional defects and five minor defects okay so here we have to calculate the probability of x equals to y equals 5 and z equals 3 so it's 10 factorial over 2 factorial 5 factorial 3 factorial times 0 .2 our 2 times 0 .5 5 times 0 .3 per 3.

03:12 So it's 0 .085...

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