Question

This question is intended to assess your understanding of the rules of probability, the Theorem of Total Probability and Bayes' formula. You should be able to answer this question after working through Section 1 of Book 1. A manufacturer obtains supplies of a particular component from three different sources: 15%, 50% and 35% of its supplies are obtained from sources 1, 2 and 3. respectively. The percentages of components from sources 1, 2 and 3 that prove to be faulty are 4%, 1% and 2%, respectively. (a) Identify the events involved by giving them names, and write down the information supplied as probabilities using these names. (For example, let E, be the event that a component comes from source 1: P(E) = ....) (b) Calculate the probability that a randomly selected component will prove to be faulty. (c) Given that a component is found to be faulty, calculate the probability that it was obtained from source 2. (d) Given that a component is not faulty, calculate the probability that it was obtained from source 3.

Answer 1

Abolet E E2, E3 be the events that the component is supplied from 1,2,oh 3 sources despechiul and het F be the event that the component is howlty So, the genen, information is. PGE) = 15% = 0.15 ; PCE ad=50% =0.50 PCE3) = 35% = 0.35... & PCE/E) = 4% =0.04, PCFLE 2d=1% =0.og APCF(EB) = 2%=0.02. aby we need to find PCF) By total Probability law on formula PCF) - PCFIE.). PCE.) +PCF/E2) PCE2) +PCFLES). PCE3).

E d.o4X0.15 + 0.01 0.5oto02x035 = 0.018 ♡ To find, PC E2/F). By bayes rule, PCERCE) - PCF/E2), PCE) PCF) PCEM (F) = 0.277 0.01x0.50 0.018 & PC Ez/E). Where Ē = event that component is not faulty, I so, from, ca, PCFIE,)=0.96, PCF (E2) = 0.99, PCELE3) = 0.98. Hence, by bayes Rule. I PCEB/F) = PCELER).P (E8) PCÊ ) = PCE (EB). P(E3) 2 As PC)=1-PCA) 1-PCF) - PG E3(E) = 0.98×0.35 . 1- 0.018 0.350