Question

The Earth has been invaded by aliens that pose as humans! Fortunately, the government has developed a test that correctly identifies allens 95% of the time. Unfortunately, 10% of the time it incorrectly identifies a human as an alien. If 2% of all humans have been replaced with aliens, find the probability that when the test Identifies a person as an alien, he actually is one. (n = 1,000)​

Answer 1

Using Bayes' theorem, the probability that a person is an alien given they tested positive by the test is approximately 16.24%.

Step-by-step explanation:

Finding the Probability That a Person Identified as an Alien Is Actually an Alien

To answer the question, we can use Bayes' theorem which gives us a way to update the probability estimates based on new information. Given that the test correctly identifies aliens 95% of the time and incorrectly identifies humans as aliens 10% of the time, combined with the information that 2% of all humans have been replaced with aliens, we can calculate the probability of a person being an alien given they tested positive for being an alien.

Let's denote:

A as the event that a person is an alien.

Not A as the event that a person is not an alien.

T as the event that the test identifies a person as an alien.

We already know that:

P(A) = 0.02 (Probability of being an alien)

P(Not A) = 0.98 (Probability of not being an alien)

P(T|A) = 0.95 (Probability that the test is positive given the person is an alien)

P(T|Not A) = 0.10 (Probability that the test is positive given the person is not an alien)

What we want to find is P(A|T), the probability that a person is an alien given that they tested positive for being an alien. Using Bayes' theorem:

P(A|T) = (P(T|A) * P(A)) / (P(T|A) * P(A) + P(T|Not A) * P(Not A))

P(A|T) = (0.95 * 0.02) / (0.95 * 0.02 + 0.10 * 0.98)

P(A|T) = (0.019) / (0.019 + 0.098)P(A|T) = 0.019 / 0.117

P(A|T) ≈ 0.1624Therefore, when the test identifies a person as an alien, there is approximately a 16.24% chance that they actually are an alien.