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5. Applying Bayes' Theorem
Bayes’ Theorem provides a formula that we can use to relate conditional probabilities. Learn how to use Bayes’ Theorem in this lesson.
Lesson Goal (00:11)
The goal of this lesson is to learn how to use Bayes’ Theorem to calculate conditional probabilities.
Bayes’ Theorem Formula (00:16)
Bayes’ Theorem is a formula for calculating conditional probabilities using several other simple probabilities.
To derive the formula, we should understand that the probability of two events A and B occurring can be the probability of A times the conditional probability of B given A, or the probability of B times the conditional probability of A given B. If we consider these two formulas to be equal, we can find a formula for the conditional probability of A given B which will include the conditional probability of B given A, as well as the probabilities of A and B. This formula can be used in a wide variety of situations.
Understanding the Problem (01:07)
In this lesson, we consider a newly developed cancer test. We know the probability that a person tests positive given that they actually have cancer, but we want to find the probability that a person actually has cancer given that they test positive.
We can use Bayes’ theorem to calculate this probability if we know the probability a person actually has cancer, the probability a person tests positive, and the conditional probability that a person tests positive given that they actually have cancer.
Calculating the Probabilities (02:36)
In our problem, we need to calculate the probability that a person tests positive for cancer. To do this, we find the probability of someone actually having cancer and testing positive given that they have cancer. We then add the probability of someone not having cancer and testing positive given that they do not have cancer.
Once we know the probability of someone testing positive, we can calculate the probability that a person actually has cancer given that they test positive. We multiply the probability of a person having cancer by the probability that they test positive given they have cancer, then divide by the probability of testing positive.
The result of this calculation depends on the parameters of the problem. In our case, the test is 99% accurate for people who have cancer, and 97% accurate for people who do not have cancer. However, we find that only 40% of people who test positive actually have cancer. This is because the form of cancer is rare, and as a result, the number of healthy people incorrectly diagnosed outnumbers the small number of people who actually have cancer.