Q: A department store finds that 30% of customers pay by credit card. For the next 5 customers, what is the probability that exactly 3 pay by credit card?

$$10 \times 0.3^3$ \times 0.7^2$

Q: Suppose you flip three coins, each showing heads with probability $$0.2. $$What is the probability of getting exactly two heads?

$$3 \times 0.2^2$ \times 0.8^1$

Q: A quality inspector tests 20 bulbs, each with a 5% chance of being defective. What is the probability that exactly 2 bulbs are defective?

$$\dfrac{20!}{2!18!} \times 0.05^2$ \times 0.95$$^{18}$$

Q: If $$X \sim \text{Bin}(3, 0.2)$$, what is $$P(X=1)$$?

$$3 \times 0.2^1$ \times 0.8^2$

Question 1

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

Accepted Answer

The number of defective items found in a sample of 20 tested products.

Question 2

Let $$X \sim 	ext{Ber}(p). $$What is the variance of $$X$$?

Accepted Answer

$$p(1-p)$$

Question 3

A factory produces light bulbs, each with a 5% chance of being defective. If a quality inspector tests 20 bulbs, what is the expected number of defective bulbs?

Accepted Answer

$$1$$

Question 4

Which formula gives the probability of observing exactly $$x$$ successes in $$n$$ independent Bernoulli trials with success probability $$p$$?

Accepted Answer

$$\dfrac{n!}{x!(n-x)!}p^x(1-p$$)^{n-x}$$

Question 5

Suppose $$X \sim 	ext{Bin}(n, p). $$Which of the following expressions correctly gives the expected value $$E[X]$$?

Accepted Answer

$$np$$

Question 6

A department store finds that 30% of customers pay by credit card. For the next 5 customers, what is the probability that exactly 3 pay by credit card?

Accepted Answer

$$10 	imes 0.3^3$ 	imes 0.7^2$

Question 7

If $$X \sim 	ext{Bin}(n, p)$$, what is the variance $$	ext{Var}(X)$$?

Accepted Answer

$$np(1-p)$$

Question 8

Which statement is TRUE about the relationship between the Bernoulli and Binomial distributions?

Accepted Answer

The Bernoulli distribution is a special case of the Binomial distribution when $$n=1.$$

Question 9

A customer randomly selects a DVD player from a shipment containing 3 defective and 7 non-defective players. What is the probability the customer selects a defective player?

Accepted Answer

$$0.3$$

Question 10

Suppose you flip three coins, each showing heads with probability $$0.2. $$What is the probability of getting exactly two heads?

Accepted Answer

$$3 	imes 0.2^2$ 	imes 0.8^1$

Question 11

A quality inspector tests 20 bulbs, each with a 5% chance of being defective. What is the probability that exactly 2 bulbs are defective?

Accepted Answer

$$\dfrac{20!}{2!18!} 	imes 0.05^2$ 	imes 0.95$$^{18}$$

Question 12

Given $$X \sim 	ext{Bin}(20, 0.05)$$, what is the probability that there are exactly 2 defective bulbs, given that there is at least one defective bulb?

Accepted Answer

$$\dfrac{P(X=2)}{1-P(X=0)}$$

Question 13

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

Accepted Answer

The probability of success changes from trial to trial.

Question 14

If $$Y_i \sim 	ext{Ber}(p)$$ are independent and identically distributed, and $$X = \sum_{i=1}^n Y_i$$, what is the distribution of $$X$$?

Accepted Answer

$$X \sim 	ext{Bin}(n, p)$$

Question 15

Which real-world scenario is best modeled by a Bernoulli distribution?

Accepted Answer

Whether a randomly selected product is defective or not.

Question 16

For $$X \sim 	ext{Bin}(n, p)$$, which of the following is the correct formula for the probability mass function (PMF)?

Accepted Answer

$$p(x) = \dfrac{n!}{x!(n-x)!} p^x (1-p$$)^{n-x}$$ for x = 0$$, 1, ..., n$$

Question 17

If $$X \sim 	ext{Bin}(3, 0.2)$$, what is $$P(X=1)$$?

Accepted Answer

$$3 	imes 0.2^1$ 	imes 0.8^2$

Question 18

Which of the following is a common application of the binomial distribution in business?

Accepted Answer

Modeling the number of defective items in a batch during quality control.

L5.1 Discrete Distributions: Bernoulli and Binomial Distributions

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