Probability and Statistics: Conditional Probability and Independence

A factory has two machines, A and B. Machine A produces 60% of the items, while Machine B produces 40%. 2% of the items produced by Machine A are defective, while 3% of the items produced by Machine B are defective. If a randomly selected item is found to be defective, what is the probability it was produced by Machine B?

A software company has two teams, Team A and Team B. Team A completes 60% of projects, and Team B completes 40%. Team A has a 90% success rate, and Team B has an 80% success rate. If a project is successful, what is the probability it was completed by Team B?

In a certain city, 30% of the population has a certain disease. A test for the disease is 90% accurate (true positive rate), and has a 10% false positive rate. If a person tests positive, what is the probability that they actually have the disease?

What's the probability that someone has a college degree given they voted?

If P(A) is 0.34, P(B) is 0.50, and P(A \cup B) is 0.70, what is the probability of A, given that B has occurred?

What is the probability of $A \cup B$, given that event $C$ occurs?

Given the cause was cardiovascular in nature, what is the probability the person was female?

What is the probability the death was cardiovascular in nature, given the person was female?

Given the cause was cerebral or respiratory in nature, what is the probability the person was male?

What is the conditional probability of $A \cap C$, given $B \cap C$?

What is the probability of $B^c$, given $A \cup C$?

Is the probability of A, given B equal to, less than, or greater than the unconditional probability of A?

If event A is a subset of event B and P(A) > 0, what can be said of P(A|B) and P(B|A)?

Given that A is a subset of B and $P(A) > 0$, what can be said about the probability of A given B?

Given the events A (Rahul eats a bagel for breakfast) and B (Rahul eats pizza for lunch) with the probabilities $P(A) = 0.6$, $P(B) = 0.5$, and the conditional probability $P(A|B) = 0.7$, find the conditional probability $P(B|A)$, rounded to the nearest hundredth.

Given a product space where $\Omega = \Omega_1 \times \Omega_2$, show that the random variables $X$ and $Y$ are independent when $X$ is a function of $\omega_1$ and $Y$ is a function of $\omega_2$.

Given two events A and B, determine if they are mutually exclusive or independent. Explain your reasoning.

What’s the probability that you’ll draw an ace, hold onto it, and then draw a king?