Explain bayes theorem of probability with example

Fresh day جدة, let's solve. Bayes' theorem is a fundamental concept in probability theory that describes how to update the probability of a hypothesis based on new evidence. It's expressed as: P(A|B) = (P(B|A)P(A))/(P(B)) Where: P(A|B) is the posterior probability*: the probability of hypothesis A given evidence B. P(B|A) is the likelihood*: the probability of evidence B given hypothesis A. P(A) is the prior probability*: the initial probability of hypothesis A before observing evidence B. P(B) is the probability of the evidence*: the overall probability of observing evidence B. Example: Suppose a medical test for a rare disease is 99% accurate (meaning it correctly identifies 99% of people who have the disease and correctly identifies 99% of people who don't have the disease). The disease affects 0.1% of the population. If a person tests positive, what is the probability they actually have the disease? Let A be the event that a person has the disease, and B be the event that the person tests positive. P(A) = 0.001 (prior probability of having the disease) P(B|A) = 0.99 (probability of testing positive given you have the disease - true positive rate) P(not A) = 0.999 (prior probability of not having the disease) P(B|not A) = 0.01 (probability of testing positive given you don't have the disease - false positive rate) We need to find P(A|B). First, calculate P(B): P(B) = P(B|A)P(A) + P(B|not A)P(not A) P(B) = (0.99 × 0.001) + (0.01 × 0.999) P(B) = 0.00099 + 0.00999 = 0.01098 Now, apply Bayes' theorem: P(A|B) = (P(B|A)P(A))/(P(B)) = (0.99 × 0.001)/(0.01098) = (0.00099)/(0.01098) ≈ 0.09016 So, even with a positive test, the probability of actually having the disease is only about 9.02%, due to the rarity of the disease. Send me the next one 📸