Answer to Question #122052 in Statistics and Probability for Ali khan

Question

A firm purchases a microchip used in the manufacturing of LED screens from four different
suppliers. TechZone supplies 30 percent of the total microchips, Advance Electronics 25 percent,
PSP importers 18 percent, and TechParts inc. 32 percent. TechZone tends to have the best
quality, as only 3 percent of its microchips arrive defective. Advance Electronics chips are 4
percent defective, PSP importers chips are 7 percent defective, and Parts Inc. are 6.5 percent
defective.
a) If a random chip is selected, what is the probability that the chip is defective
b) A defective chip was discovered in the latest shipment. What is the probability that it came
from TechZone supplies?
c) A defective chip was discovered in the latest shipment. What is the probability that it came
from Parts Inc.?

Answer 1

Let:

$A$ denote the event that chip came from TechZone,

$B$ denote the event that chip came from Advance Electronics,

$C$ denote the event that chip came from PSP,

$F$ denote the event that chip came from Parts Inc, and

$D$ denote the event that the randomly selected chip is defective.

Given

$P(A)=0.30, P(D|A)=0.03,$

$P(B)=0.25, P(D|B)=0.04,$

$P(C)=0.18, P(D|C)=0.07,$

$P(F)=0.32,P(D|F)=0.065.$

a) If a random chip is selected, what is the probability that the chip is defective

The total probability rule

$$P(D)=P(A)P(D|A)+P(B)P(D|B)+$$ $$+P(C)P(D|C)+P(F)P(D|F)=$$ $$=0.3(0.03)+0.25(0.04)+$$ $$+0.18(0.07)+0.32(0.065)=0.0524$$

b) A defective chip was discovered in the latest shipment. What is the probability that it came from TechZone supplies?

We apply Bayes' Theorem to compute this probability

$$P(A|D)=\dfrac{P(A)P(D|A)}{P(D)}=\dfrac{0.3(0.03)}{0.0524}=\dfrac{45}{262}\approx$$ $$\approx0.1718$$

c) A defective chip was discovered in the latest shipment. What is the probability that it came from Parts Inc.?

We apply Bayes' Theorem to compute this probability

$$P(F|D)=\dfrac{P(F)P(D|F)}{P(D)}=\dfrac{0.32(0.065)}{0.0524}=\dfrac{52}{131}\approx$$ $$\approx0.3969$$