Q: A basketball player takes 12 free throws, and the probability of making each shot is 0.7. What is the expected number of successful free throws?

$$E[X] = n \times p = 12 \times 0.7 = 8.4$$

Q: Which formula gives the variance of a binomial random variable $$X$$ with parameters $$n$$ and $$p$$?

$$Var(X) = n \times p \times (1-p)$$

Question 1

Which of the following is NOT an assumption of the binomial model?

Accepted Answer

The probability of success changes from trial to trial.

Question 2

A basketball player takes 12 free throws, and the probability of making each shot is 0.7. What is the expected number of successful free throws?

Accepted Answer

$$E[X] = n 	imes p = 12 	imes 0.7 = 8.4$$

Question 3

Which formula gives the variance of a binomial random variable $$X$$ with parameters $$n$$ and $$p$$?

Accepted Answer

$$Var(X) = n 	imes p 	imes (1-p)$$

Question 4

A company receives an average of 5 customer emails per hour. What is the probability that exactly 3 emails are received in a given hour, assuming a Poisson process?

Accepted Answer

$$P(X=3) = \frac{5^3$ e$$^{-5}$$}{3!}$$

Question 5

Which of the following scenarios is best modeled by a Poisson distribution?

Accepted Answer

The number of cars passing through a toll booth in one hour.

Question 6

If you sample 8% of a large population without replacement, which condition allows you to use the binomial model as an approximation?

Accepted Answer

The 10% condition: the sample is less than 10% of the population.

Question 7

Which of the following is the correct probability mass function for a binomial random variable $$X$$ with parameters $$n$$ and $$p$$?

Accepted Answer

$$P(X = k) = \binom{n}{k} p^k (1-p$$)^{n-k}$$

Question 8

A call center receives an average of 10 calls per hour. What is the expected number and variance of calls in 2 hours, assuming a Poisson process?

Accepted Answer

Expected value: $$20$$, Variance: $$20$$

Question 9

Which of the following statements about the Poisson distribution is TRUE?

Accepted Answer

The mean and variance of a Poisson random variable are equal.

Question 10

Suppose you flip a fair coin 20 times. What is the probability of getting exactly 10 heads?

Accepted Answer

$$P(X=10) = \binom{20}{10} (0.5$$)^{10}$$ (0.5$$)^{10}$$

Question 11

A binomial experiment consists of 15 trials with a probability of success $$p = 0.2. $$What is the standard deviation of the number of successes?

Accepted Answer

$$\sqrt{15 	imes 0.2 	imes 0.8}$$

Question 12

Which of the following is a key difference between the binomial and Poisson models?

Accepted Answer

The binomial model has a fixed number of trials, while the Poisson model counts events in a fixed interval.

Question 13

A quality control inspector checks 100 products, each with a 2% chance of being defective. What is the expected number of defective products?

Accepted Answer

$$E[X] = 100 	imes 0.02 = 2$$

Question 14

If the probability of receiving an email in a minute is very small, but there are many minutes in an hour, which model is most appropriate for the number of emails received in an hour?

Accepted Answer

Poisson model

Question 15

Which of the following is the correct probability mass function for a Poisson random variable $$X$$ with mean $$\lambda$$?

Accepted Answer

$$P(X = k) = \frac{\lambda^k e$$^{-\lambda}$$}{k!}$$

Question 16

A binomial random variable $$X$$ has $$n = 50$$ and $$p = 0.1. $$What is the variance of $$X$$?

Accepted Answer

$$Var(X) = 50 	imes 0.1 	imes 0.9 = 4.5$$

Question 17

Which of the following is NOT a property of a Poisson process?

Accepted Answer

There is a fixed number of trials.

Question 18

A store receives an average of 3 customers per hour. What is the probability that no customers arrive in a given hour?

Accepted Answer

$$P(X=0) = \frac{3^0$ e$$^{-3}$$}{0!} = e$$^{-3}$$

Question 19

Which of the following best describes the random variable in a binomial model?

Accepted Answer

The number of successes in $$n$$ independent Bernoulli trials.

Probability Models for Counts: Binomial and Poisson Distributions

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