Conditional probability and Bayes’ theorem：練習問題

In a bag, there are 10 red balls and 8 blue balls. 4 of the red balls have stripes, and 3 of the blue balls have stripes. If a ball is randomly selected from the bag and found to have stripes, what is the probability that it is a red ball?

A screening test for a certain disease has a sensitivity of 99% and a specificity of 95%. In a population where 2% of the people have the disease, what is the probability that a person who tests positive actually has the disease? Give your answer correct to 4 decimal places.

A factory produces parts using three shifts: Morning (M), Afternoon (A), and Night (N). The Morning shift produces 50% of the parts, the Afternoon shift produces 30%, and the Night shift produces 20%. The probabilities that a part produced by these shifts is defective are 0.02, 0.03, and 0.05, respectively. If a randomly chosen part is found to be defective, and it is known that this defective part was not produced by the Morning shift, what is the probability that it was produced by the Night shift?

In a factory, Machine 1 (M1) produces 70% of all items and Machine 2 (M2) produces the remaining 30%. The probability that an item produced by M1 is defective (D) is 0.05, and the probability that an item produced by M2 is defective is 0.10. If a randomly selected item is found to be non-defective (D'), what is the probability that it was produced by Machine 1?

Event A has probability \(P(A) = 0.4\). Event B has probability \(P(B) = 0.7\). The probability that both A and B occur is \(P(A \cap B) = 0.3\). Find the probability that event A occurs given that event B does not occur, i.e., \(P(A|B')\).

A fair six-sided die is rolled. Let A be the event that the number rolled is odd, and B be the event that the number rolled is greater than 3. Find the probability of A given B, i.e., $$P(A|B)$$ .

A company manufactures computer chips using two production lines, L₁ and L₂. It is known that L₁ produces 60% of the total output and L₂ produces 40%. The probability that a chip produced by L₁ is defective is 0.05, while the probability that a chip produced by L₂ is defective is 0.10. If a chip is randomly selected and found to be defective, find the probability that it was produced by L₁.

Box A contains 4 red balls and 6 blue balls. Box B contains 7 red balls and 3 blue balls. A fair coin is tossed. If it lands heads, Box A is chosen; if it lands tails, Box B is chosen. A ball is then drawn at random from the chosen box. If the ball drawn is red, what is the probability that it came from Box B?

An online clothing store analyzes its sales data and finds that \(60\%\) of customers pay by credit card, while the remaining \(40\%\) use an e-wallet. The return rates for items paid by credit card and e-wallet are \(5\%\) and \(8\%\) respectively.

(a) Find the probability that a randomly selected item is returned.
(b) Given that an item is returned, find the probability that it was paid by an e-wallet.
(c) Suppose two items are selected at random. Given that both items are returned, find the probability that both were paid by credit cards.

An insurance company classifies its policyholders into three risk categories: Low, Medium, and High. The distribution of policyholders and the probability of filing at least one claim in a year for each category are shown below:

(a) Find the probability that a randomly selected policyholder files a claim in a year.
(b) Given that a policyholder files a claim, what is the probability that they belong to the "High" risk category?
(c) Suppose a policyholder is randomly selected. If this person filed a claim in the first year, find the probability that they will also file a claim in the second year, assuming their risk category remains the same and claims in different years are independent.