A defibrillator manufacturer uses three different materials (Silver, Titanium, a

Question

A defibrillator manufacturer uses three different materials (Silver, Titanium, and Copper) for the electrodes in its devices. Each electrode is made solely from a single material. The electrodes are designed as an adhesive pad, which must carry strong pulses of electricity without deteriorating and remain functional after 10 years. Events are defined as follows: .Ag: the electrode is made of Silver Ti: the electrode is made of Titanium . Cu: the electrode is made of Copper . X: the electrode is functional after 10 years The probabilities that an electrode is made out of the different materials are as follows: P(Ag) 0.2, P(Ti) 0.7, P(Cu) 0.1 and the conditional probabilities of functionality after 10 years are as follows: P(XIAg) = 0.9, P(XlTi) = 0.8, P(XCu) = 0.6 i. Using words, explain what P(Ag|X) represents. Then, compute P(Ag|X). ii. Using words, explain what PGAg,IX) represents. Then, compute P(Ag,IX). iii. Using words, explain what P(X) represents. Then, compute P(X).

Explanation / Answer

i) P(Ag/X) is represent the probabiltiy that functionality after 10 to the electorde is made of Silver.
By using Bayes theorem
P(Ag/x) = P(x/Ag) P(Ag) / [ P(x/Ag) P(Ag) + P(x/Ti) P(Ti) + P(x/Cu) P(Cu)]
= (0.9*0.2) / [0.9*0.2 + 0.8*0.7 + 0.1*0.6]
= 0.18 / 0.8 = 0.225

ii) P(Ag'/X) is represent the probabiltiy that functionality after 10 to the electorde is not made of Silver.
By using Bayes theorem
P(Ag'/x) = 1 - P(Ag/x) = 1 - 0.225 = 0.775

iii)
P(x) is represent is the probabiltiy that functionality after 10 is
P(x) = P(x/Ag) P(Ag) + P(x/Ti) P(Ti) + P(x/Cu) P(Cu)
= 0.9*0.2 + 0.8*0.7 + 0.1*0.6 = 0.8

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