Challenge 2: Bayes' Theorem
In the world of probability and statistics, Bayesian thinking offers a framework for understanding the probability of an event based on prior knowledge. It contrasts with the frequentist approach, which determines probabilities based on the long-run frequencies of events. Bayes' theorem is a fundamental tool within this Bayesian framework, connecting prior probabilities and observed data.
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Imagine you are a data scientist working for a medical diagnostics company. Your company has developed a new test for a rare disease. The prevalence of this disease in the general population is 1%. The test has a 99% true positive rate (sensitivity) and a 98% true negative rate (specificity).
Your task is to compute the probability that a person who tests positive actually has the disease.
Given:
- P(Disease) = Probability of having the disease =
0.01 - P(Positive|Disease) = Probability of testing positive given that you have the disease =
0.99 - P(Negative|No\ Disease) = Probability of testing negative given that you don't have the disease =
0.98
Using Bayes' theorem:
P(Disease|Positive) = P(Positive|Disease) * P(Disease) / P(Positive)
Where P(Positive) can be found using the law of total probability:
P(Positive) = P(Positive|Disease) * P(Disease) + P(Positive|No Disease) * P(No Disease)
Compute P(Disease|Positive), the probability that a person who tests positive actually has the disease.
Soluzione
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