Probability Questions

The time until a computer system crashes is exponentially distributed with a mean of 2500 hours. What is the probability that the system will crash within the first 1000 hours?

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?

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?

The time it takes for a certain type of light bulb to burn out is exponentially distributed with a mean of 500 hours. What is the probability that a randomly selected bulb will last longer than 1000 hours?

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?

You roll two six-sided dice. Are the events "getting a sum of 7" and "getting doubles" independent events?

A company's daily stock price fluctuations are approximately normally distributed with a mean of $0 (no change) and a standard deviation of $2. What is the probability that the stock price will increase by more than $4 on a given day?

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?

The lifetime of a certain battery follows a gamma distribution with a shape parameter of 2 and a scale parameter of 500 hours. What is the average lifetime of this battery?

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