ull tile Saple space. Indicate events A, B, C, AC, AB, and AUB on the sample spa

Question

ull tile Saple space. Indicate events A, B, C, AC, AB, and AUB on the sample space; b) Find P(AB), PA), P(BUÁ), PIA1B), and P(BIA) 2. Consider the possible failure of water supply systems to meet the demand during any given summer day. Three demand levels could happen: demand level of 100,000 gal/day; P(D) = 0.6 demand level of 150,000 gal/day; P(D) = 0.3 Da demand level of 200,000 gal/day; P(D) = 0.1 If let E denotes as the failure to meet the supply, then it is given that P(E|D) = 0, P(El D2-0.1, and P(EIDs) = 0.5. Use total probability to determine the probability that the supply will be insufficient if the pro a) b) Find the probability that a demand level of 150,000 gal/day was the 'cause' of the system's failure to meet demand if an inadequate supply was observed (using Bayes theorem). 3. Computer chips often contain surface imperfections. For a certain type of computer chip, the probability mass function of the number of defects X is presented in the following table

Explanation / Answer

1. From total probability theorem,

P(E) = P(E|D1) * P(D1) + P(E|D2) * P(D2) + P(E|D3) * P(D3)

= = 0*0.6 + 0.1*0.3 + 0.5*0.1

= 0.08

2. From Bayes' theorem.

P(D2 | E) =  P(E|D2) * P(D2) / P(E) = 0.1*0.3 / 0.08 = 3/8 = 0.375

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