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A quality control inspector tests batches of 20 items. If the probability of a single item being defective is 0.05, what is the probability of finding exactly 2 defective items in a batch using the binomial distribution?

Calls arrive at a customer service center at an average rate of 4 calls per hour. What is the probability of receiving exactly 3 calls in a given hour using the Poisson distribution?

A baseball player has a 0.300 batting average. What is the probability they will get their 5th hit on their 15th at-bat using the negative binomial distribution?

In a geometric distribution with success probability 0.2, what is the probability that the first success occurs on the 4th trial?

From a deck of 52 cards, 7 cards are drawn without replacement. What is the probability of getting exactly 3 hearts using the hypergeometric distribution?

In a multinomial experiment with n=10 trials and probabilities θ₁=0.5, θ₂=0.3, θ₃=0.2, what is the probability of getting x₁=5, x₂=3, x₃=2?

A jar contains 8 red, 5 blue, and 7 green marbles. If 6 marbles are drawn without replacement, what is the probability of getting 2 red, 2 blue, and 2 green marbles using the multivariate hypergeometric distribution?

For a binomial distribution with n=50 and θ=0.4, what are the approximate mean and standard deviation using the normal approximation?

If X follows a Poisson distribution with λ=6, what is P(X ≤ 3)?

The number of trials needed to get 3 successes with success probability 0.25 follows a negative binomial distribution. What is the expected number of trials needed?

For a geometric distribution with p=0.1, what is the probability that more than 8 trials are needed for the first success?

A lot of 100 items contains 10 defectives. If a sample of 8 items is selected without replacement, what is the expected number of defectives in the sample?

In a binomial experiment with n=100 trials, what is the minimum success probability needed for the normal approximation to be reasonable?

If accidents occur at a factory at an average rate of 2 per week, what is the probability of no accidents occurring in a given week?

A basketball player makes 80% of free throws. What is the probability they make their 10th successful free throw on their 12th attempt?

For a geometric distribution with mean 5, what is the variance?

From a population of 50 items with 15 defectives, a sample of 10 is drawn without replacement. What is the variance of the number of defectives in the sample?

In a trinomial experiment with n=20 trials and probabilities (0.5, 0.3, 0.2), what is the covariance between X₁ and X₂?

If X ~ Poisson(λ=5), what is P(3 ≤ X ≤ 7)?

A fair coin is tossed until the 3rd head appears. What is the probability that exactly 7 tosses are needed?

For a geometric distribution with p=0.1, what is the probability that the first success occurs within the first 5 trials?

A box contains 12 red and 8 blue balls. If 5 balls are drawn without replacement, what is the probability of getting at least 3 red balls?

In a binomial experiment, if n=25 and θ=0.4, what is P(8 ≤ X ≤ 12) using normal approximation?

If cars arrive at a toll booth at an average rate of 10 cars per minute, what is the probability of more than 12 cars arriving in a minute?

A student takes a 20-question multiple choice test with 4 choices each. If they guess randomly, what is the probability of getting exactly 8 correct answers?

For a negative binomial distribution with r=4 and p=0.3, what is the variance?

In a hypergeometric distribution with N=100, n=10, k=20, what is the probability of exactly 3 successes?

A multinomial experiment has 50 trials with probabilities (0.4, 0.3, 0.2, 0.1). What is the expected value of X₂?

If X follows a geometric distribution with mean 8, what is P(X = 6)?

For a Poisson distribution with λ=8, what is the probability of exactly 10 events?

In a negative binomial experiment with r=3 and p=0.4, what is the probability that exactly 7 trials are needed?

A box contains 15 good and 5 defective items. If 4 items are drawn without replacement, what is the probability of at most 1 defective?

For a binomial distribution with n=40 and θ=0.35, what is the z-score for x=16 using normal approximation?

If emails arrive at an average rate of 2 per hour, what is the probability of receiving no emails in a 2-hour period?

A geometric distribution has variance 20. What is the probability of success on a single trial?

In a hypergeometric distribution with N=50, n=8, k=15, what is the expected value?

A multinomial experiment has 30 trials with probabilities (0.5, 0.3, 0.2). What is the variance of X₁?

For a Poisson distribution, if P(X=2) = P(X=3), what is λ?

A negative binomial distribution has mean 10 and variance 30. What are the values of r and p?

In a geometric distribution, if the probability of success on the first trial is 0.2, what is the probability that the first success occurs on an odd-numbered trial?

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