貝葉斯定理：練習題

A diagnostic test for a rare disease is known to be \(95\%\) accurate for those who have the disease (sensitivity) and \(90\%\) accurate for those who do not (specificity). If \(1\%\) of the population actually has the disease, find the probability that a person who tests positive actually has the disease.

A factory uses three shifts to produce light bulbs: Morning, Afternoon, and Night. The Morning shift produces 50%, the Afternoon 30%, and the Night 20% of the total output. The probability of a defective bulb is 0.01 for the Morning shift, 0.02 for the Afternoon shift, and 0.05 for the Night shift. If a bulb is selected at random and found to be defective, calculate the probability it was produced during the Night shift.

An email filter classifies messages as 'Spam' or 'Ham'. It is known that 70% of all emails are Spam. The word "FREE" occurs in 60% of Spam emails and 5% of Ham emails. If an email contains the word "FREE", determine the probability that it is actually Spam.

A company filters spam emails using a software that identifies 98% of spam as such, but misidentifies 3% of legitimate emails as spam. If 80% of all incoming emails are spam, find the probability that an email identified as spam is actually legitimate.

A weather station predicts rain based on current atmospheric conditions. Historically, it rains on 20% of days. When it rains, the station correctly predicts rain 85% of the time. When it does not rain, the station incorrectly predicts rain 10% of the time.

(a) Find the probability that the station predicts rain on any given day.
(b) If the station predicts rain, calculate the probability that it actually rains.

A bank uses an automated system to classify loan applications as 'Low Risk', 'Medium Risk', or 'High Risk'. 60% of applications are Low Risk, 30% are Medium Risk, and 10% are High Risk. Historical data shows that 1% of Low Risk, 7% of Medium Risk, and 25% of High Risk loans eventually default.

(a) Calculate the probability that a randomly selected loan will default.
(b) A loan has defaulted. Calculate the probability it was originally classified as 'Medium Risk'.
(c) The bank manager claims that if a loan defaults, it is more likely to have come from the 'High Risk' category than the 'Low Risk' category. Verify whether this claim is correct.