Probability Questions

You are simulating a random process with a known theoretical mean. After many simulations, you find that the average of your simulated results is significantly different from the expected theoretical mean. What is the MOST likely explanation for this discrepancy?

What does a conditional probability P(Y=y | X=x) represent in the context of joint distributions?

Which of the following is NOT an assumption or condition that should be met when applying the Central Limit Theorem?

If the time to complete a task is exponentially distributed with a mean of 20 minutes, what is the probability that the task takes less than 10 minutes?

Events A and B are independent. The probability of event A is 0.3. The probability of event B is 0.6. What is the probability of both A and B occurring?

A machine produces widgets with a defect rate of 5%. What is the probability that a batch of 20 widgets will contain exactly 2 defective ones?

The Central Limit Theorem states that the sampling distribution of the sample mean will approach what distribution as the sample size increases, regardless of the shape of the population distribution?

A company produces light bulbs with a lifespan that is skewed to the right. The average lifespan is 1000 hours with a standard deviation of 100 hours. If you randomly select 100 light bulbs, what is the approximate probability that their average lifespan is greater than 1010 hours?

The weight of apples is normally distributed with a mean of 150g and a standard deviation of 10g. What is the z-score for an apple weighing 165g?

Suppose a diagnostic test for a certain disease has a 95% sensitivity and 90% specificity. It means that the test correctly identifies 95% of people with the disease and correctly identifies 90% of people without the disease. If the prevalence of the disease in a population is 1%, what is the probability that a person who tests positive actually has the disease?