Conditional Probability and Bayes' Theorem

When is the conditional probability of event E given event F defined as $P(E|F) = \frac{P(E \cap F)}{P(F)}$?

Given that events E1, E2,... are mutually exclusive, what does $P(\bigcup_i E_i | F)$ equal?

Which of the following is a correct application of the multiplication rule for conditional probabilities?

An urn contains 4 red balls and 6 blue balls. Two balls are drawn without replacement. What is the probability that the first ball is red and the second is blue?

What does the Law of Total Probability state for events E and F?

A company estimates that 20% of the population is 'tech-savvy' and will adopt their new gadget. Tech-savvy people will buy the gadget with a probability of 0.6. Non-tech-savvy people will buy it with a probability of 0.1. What is the probability that a randomly selected person will buy the gadget?

A factory has two machines, A and B, that produce bolts. Machine A produces 60% of the bolts, and 5% of its bolts are defective. Machine B produces 40% of the bolts, and 10% of its bolts are defective. If a bolt is randomly selected and found to be defective, what is the probability that it was produced by machine A?

For events E and F, when are they considered independent?

If events A and B are independent, which of the following statements is true?

If events E and F are mutually exclusive and $P(E) > 0$ and $P(F) > 0$, are they independent?

What condition defines three events E, F, and G as mutually independent?

If A, B, and C are events and $P(C) > 0$, when are A and B considered conditionally independent given C?

Which of the following is an equivalent condition for conditional independence of events A and B given event C?

A bag contains two coins: one fair and one with two heads. A coin is chosen at random and flipped twice. Both flips result in heads. Are the events 'first flip is heads' and 'second flip is heads' independent?

What is the purpose of Bayes' Theorem?

A disease affects 1% of the population. A test for the disease has a 95% sensitivity (true positive rate) and a 5% false positive rate. If a person tests positive, what is the approximate probability that they actually have the disease?

Given events A and B, and knowing P(A|B), which theorem allows us to find P(B|A)?

A weather forecaster says there is a 60% chance of rain tomorrow. Historical data shows that the forecaster is correct 80% of the time when rain is predicted, and correct 70% of the time when no rain is predicted. What is the probability that it will actually rain tomorrow?

Event A is 'a student studies' and event B is 'a student passes an exam'. If P(A) = 0.7, P(B|A) = 0.8, and P(B|A') = 0.3, what is the probability that a student passes the exam?

Suppose E and F are independent events with P(E) = 0.4 and P(F) = 0.6. What is P(E ∪ F)?

When using the Law of Total Probability, the events on which you condition must:

Two cards are drawn without replacement from a standard deck of 52 cards. What is the probability that the second card is a king, given that the first card was a king?

A coin is flipped twice. Let E be the event 'first flip is heads' and F be the event 'both flips are the same'. Are E and F independent?

A study shows that 60% of students at a university are female. It also shows that 20% of the male students and 5% of the female students are engineering majors. What percentage of all students are engineering majors?

A box contains 5 red balls and 5 blue balls. Two balls are drawn at random without replacement. What is the probability that the first ball is red and the second ball is blue?

A fair die is rolled twice. Let A be the event that the first roll is a 4. Let B be the event that the sum of the two rolls is 7. Are events A and B independent?

In a certain population, 4% of people have disease X. A new test for disease X is developed that has a 90% chance of detecting the disease if it is present and a 10% chance of producing a false positive result if the disease is not present. If a person selected at random tests positive for the disease, what is the probability that the person actually has the disease?

Events A and B are conditionally independent given C. If P(A|C) = 0.5 and P(B|C) = 0.3, what is P(A∩B|C)?

A jar contains 3 white marbles and 2 black marbles. Two marbles are drawn without replacement. What is the probability that the second marble is black given that the first marble was white?

Assume independent trials, each with a probability of success p, where 0 < p < 1. What is the probability that at least one of n trials succeeds?

Events D and E are independent; $P(D)=0.6$ and $P(E)=0.4 $. Find $P(D\mid E)$.

Urn D contains 7 white balls and 3 black balls. Urn E contains 5 white balls and 5 black balls. A fair coin is flipped: If the outcome is heads, a ball is randomly drawn from Urn D; otherwise, a ball is randomly drawn from Urn E. Suppose that a white ball is selected. What is the probability that the coin landed heads (a ball was extracted from Urn D)?