What is the probability that the test result is correct?

Question

A device tests whether each light on a string of LED lights is functioning properly. A company estimates that 2.3% of LED lights that they produce are defective. The device is 96% accurate for lights that are functioning properly and 94% accurate for lights that are defective. You use the device to test a randomly selected light. What is the probability that the test result is correct? Round your answer to the nearest tenth.

Answer 1

Let D+ denote "defective lights" and D- denote "properly functioning lights". From the provided information, we can deduce the following initial probabilities:

P(D+) = 0.023

P(D-) = 1- P(D+) = 1 - 0.023 = 0.977

P(test correct | D-) = 0.96

P(test correct | D+) = 0.94

From the preceding initial probabilities and the Law of Total Probability, we calculate that the probability our test result is correct for our randomly selected light is

P(test correct) = P(test correct and D-) + P(test correct and D+)

= P(test correct | D-)*P(D-) + P(test correct | D+)*P(D+)

= 0.96*0.977 + 0.94*0.023

= 0.95954

\approx 1.0