Probability is a numerical representation of how likely an event is to occur.

A random experiment is any experiment whose outcome cannot be predicted. The set of all possible outcomes of a random experiment is called the sample space, and an event is any subset of the sample space.

Simply put, given a random experiment and two events from it, say A and B, we say that the probability of event A occurring, given that we are already certain event B occurs, is a conditional probability.

Intuition and Probability

In some cases, especially those involving conditional probability, calculations may contradict our intuition. Here is an example of such a situation:

In a country, 10% of the population is carrying a virus. A test to detect the presence of the virus has 90% accuracy when applied to carriers and 80% accuracy when applied to non-carriers. What percentage of people actually carry the virus among those whom the test classified as carriers?

Let us see a solution that does not cite probability theorems.

Assume that the test was applied to the N inhabitants of the country. The number of tests that indicated the presence of the virus was:

where the first portion represents the 90% who are actually carriers and the second portion represents the 20% who are not carriers. Therefore, of the total, 0.27 N, 0.09 N are carriers.

So, 0.09 N / 0.27 N = 1/3 ≈ 33.3% of the people whom the test classified as carriers are actually carriers of the virus.

This number shows that a person who has taken the test and been classified as a carrier has a high chance of being a false positive. Typically, when a person takes a test of this type and the result is positive, doctors recommend a new test.