1. Answer the following questions (a) State the Total Probability Rule (b) Infor
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1. Answer the following questions (a) State the Total Probability Rule (b) Information about product failure based on chip manufacturing process contamination is given as ProbabilityLevel of Probability of Failure Contamination of Level 0.100 0.010 0.001 Medium Low 0.2 0.3 0.5 That is: there are three levels of contamination: high, medium, and low. The probability of high level of contamination is 0.2, of medium level is 0.3, of low level is 0.5. If the level is high, probability of product failure is 0.1; if the level is medium, probability of produce failure is 0.01; if the level is low, probability of product failure is 0.001. Now use the Total Probability Rule to find the probability of product failure (c) State the Bayes Theorem (d) Given the information provided in (b), let F denote the event that a product fails, and H denote the event that the product is exposed to high levels of contamination. Find P(HF)Explanation / Answer
Question 1
(a)
The total probability rule (also called the Law of Total Probability) breaks up probability calculations into distinct parts. It’s used to find the probability of an event, A, when you don’t know enough about A’s probabilities to calculate it directly. Instead, you take a related event, B, and use that to calculate the probability for A.
The probability for a can be written as sums of event B. The total probability rule is:
P(A) = P(A B) + P(A BC)
(b)
Pr(Product failure) = 0.2 * 0.100 + 0.3 * 0.010 + 0.5 * 0.001 = 0.0235
(c) Bayes theorem : a theorem describing how the conditional probability of each of a set of possible causes for a given observed outcome can be computed from knowledge of the probability of each cause and the conditional probability of the outcome of each cause.
(d)
Pr(H l F) = (0.1000* 0.2)/ (0.1000 * 0.2 + 0.01 * 0.3 + 0.001 * 0.5) = 0.851
(2) Here p for any relays functioning = 0.95
Design one probability
P(A to B) = Pr( 1 and 2 in parellel) * P(3 and 4 in parellel)
= [ 1- (1 - 0.95)2] * [1 - (1 - 0.95)2]
= 0.995
Design (2)
Pr(A to B) = Pr(1 and 3 in series) and Pr(2 and 4 are in series)
Pr(1 and 3 in series) = 0.95 * 0.95 = 0.9025
Pr(2 and 4 in series) = 0.95 * 0.95 = 0.9025
Pr(A to B) = 1 - (1 - 0.9025)2 = 0.9905
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