3. An automotive parts manufacturer reports that 45% of their product failures a

Question

3. An automotive parts manufacturer reports that 45% of their product failures are caused by assembly defects, 25% by electrical components, and 30% by mechanical defects. For this problem, suppose that ten boards fail independently. Define the random variables X, Y, and Z as the number of assembly, electrical, and mechanical defects, respectively, among those ten boards. Use this information to calculate the probabilities below. This problem asks you to apply concepts from the binomial and the multinomial distributions. a. P(X-5,Y-3,2-2) b. PX- 8)

Explanation / Answer

a)P(x=5,Y=3,Z=2) = 10! / (5!*3!*2!) * 0.45^5 * 0.25^3 * 0.30^2 = 0.0653

b) P(X=8) = P(X=8,Y=2)+P(X=8,Y=1,Z=1)+P(X=8,Z=2) = 10!/8!/2! * (0.45)^8 * 0.25^2 +10!/8! * (0.45)^8 * 0.25*0.30 +10!/8!/2! * (0.45)^8 * 0.30^2 =0.004729+0.01135+0.00681=0.02289

c)P(X=8/Y=1)= P( X=8 & Y=1) /P(Y=1)= 10!/ 8! * 0.45^8 *0.25*0.30 / [ 10!/9! * 0.75^9 *0.25] =0.01135 /0.1877=0.0604

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