Problem Sets: Conditional Probability, Bayes Theorem (条件概率、贝叶斯定理)

Suppose that \( \Pr(B) = 0.7 \), \( \Pr(A) = 0.2 \), and \( \Pr(A \cup B) = 0.8 \). What is the probability of \( \Pr(B \mid A) \)?

• If you pick a random contestant, what is the probability that she was bribing the judges?
• If you pick a random contestant, what is the probability that she was not bribing the judges?
• If you pick a random contestant, what is the probability that she is allowed to stay in the first episode?
• If you pick a random contestant, what is the probability that she is kicked off in the first episode?
• If you pick a random contestant who was allowed to stay during the first episode, what is the probability that she was bribing the judges?
• If you pick a random contestant who was kicked off during the first episode, what is the probability that she was bribing the judges?

Suppose that in a calculus class, 45% of the students are majoring in mechanical engineering, 30% in electrical engineering, and 25% in civil engineering. In the final exam, 20% of students major in mechanical engineering, 25% of students majoring in electrical engineering, and 10% of students majoring in civil engineering got a grade A.

• Randomly select 10 students who major in civil engineering. What is the probability that more than one student got a grade A. (Hint: \( 0.9^{9} \approx 0.3487, 0.9^{10} \approx 0.3487 \))

• Select a student randomly from this class. What is the probability that this student got an A in the final exam?

• Select a student who got an A at random. What is the most likely major that this student takes?

At a certain gas station, 40% of the customers use regular unleaded gas (\( A_1 \)), 35% use extra unleaded gas (\( A_2 \)), and 25% use premium unleaded gas (\( A_3 \)). Let \( B \) be the event that a customer fills up their tank with gas. Suppose that of those customers using regular gas, only 30% fill up their tank; of those using extra gas, 60% fill up their tanks, whereas of those using premium, 50% fill up their tanks.

• What is the probability that the next customer will request extra unleaded gas and fill up the tank?

• What is the probability that the next customer fills up the tank?

• If the next customer fills up the tank, what is the probability that regular gas is requested? Extra gas? Premium gas?

• Suppose that 70% of all regular fill-up customers use a credit card, 50% of all regular non-fill-up customers use a credit card, 60% of all extra fill-up customers use a credit card, 50% of all extra non-fill-up customers use a credit card, 50% of all premium fill-up customers use a credit card, and 40% of all premium non-fill-up customers use a credit card. What is the probability that the next customer will use a credit card (event \( C \))?

There are two coins in a box. One coin is fair, but the other is an unfair coin with a probability of 1/3 of heads. Select a coin randomly from the box and toss it twice. If heads are obtained on both tosses, what is the probability that the selected coin is the fair one?

One product is missing from a box containing 10 pieces of a certain product (including 5 first-class products, 3 second-class products, and 2 third-class products), and we don’t know what class it is. Now randomly pick any 2 products from the box, and the result shows that both are first-class products. What is the probability that the losing one is also first-class?.

Suppose that one virus can cause a serious disease. For all the people, there is a probability of 1% that he/she got the virus, which means he/she has a probability of 99% to be a healthy person. Suppose there is a laboratory test that can accurately detect the virus (the person carrying the virus has a probability of 100% to have a positive test result). However, the false positive rate of the test is 5%, which means that a healthy person who receives the test has a 5% chance of being misdiagnosed as a carrier of the virus.

• Randomly select 10 persons from the population. What is the probability that more than one person got the virus? (Hint: \( 0.99^8 \approx 0.9227, 0.99^9 \approx 0.9135, 0.99^{10} \approx 0.9044 \))

• Randomly select 10000 persons from the population, using the central limit theorem to approximate the probability of more than 130 persons getting the virus. (Hint: \( \Phi(1) = 0.8413, \Phi(2) = 0.9772, \Phi(3) = 0.9987, \sqrt{99} \approx 10 \))

• Select a person randomly to do the virus test. What is the probability that the test result is positive?

• Suppose a randomly selected person is tested and the test result is positive. What is the probability that this person is carrying the virus?