Q: Given the following probability distribution for the number of kids per household: $$\begin{align*} X &: 0, 1, 2 \\ P(X) &: 0.15, 0.60, 0.25 \end{align*}$$ What is the expected value (mean) of $$X$$?

$$0.15 \times 0 + 0.60 \times 1 + 0.25 \times 2 = 1.10$$

Q: A company tracks the number of complaints received daily, with the following probability distribution: $$\begin{align*} X &: 0, 1, 2, 3 \\ P(X) &: 0.45, 0.30, 0.20, 0.05 \end{align*}$$ What is the variance of $$X$$?

$$\sum (x - \mu)^2$ P(x)$$ where $$\mu = 0 \times 0.45 + 1 \times 0.30 + 2 \times 0.20 + 3 \times 0.05 = 0.85$$, \newline \text{Variance} = $$(0-0.85)^2$$ \times 0.45 + $$(1-0.85)^2$$ \times 0.30 + $$(2-0.85)^2$$ \times 0.20 + $$(3-0.85)^2$$ \times 0.05 = 0.7275$

Q: What is the probability of getting exactly 3 red marbles in 4 draws with replacement from a bag where the probability of red is 0.8?

$$\binom{4}{3} (0.8)^3$ (0.2)^1$ = 4 \times 0.512 \times 0.2 = 0.4096$$

Q: A company offers a new product, and 60% of customers are expected to purchase it after a demonstration. If you randomly select 10 customers, what is the expected number of purchases?

$$\mu = n p = 10 \times 0.6 = 6$$

Question 1

Which of the following is a discrete random variable?

Accepted Answer

The number of defective lightbulbs from a randomly chosen batch in a factory

Question 2

A probability distribution for a discrete random variable $$X$$ must satisfy which two criteria?

Accepted Answer

Each probability must be between 0 and 1, and the sum of all probabilities must be 1

Question 3

Given the following probability distribution for the number of kids per household:

$$\begin{align*} X &: 0, 1, 2 \ P(X) &: 0.15, 0.60, 0.25 \end{align*}$$

What is the expected value (mean) of $$X$$?

Accepted Answer

$$0.15 	imes 0 + 0.60 	imes 1 + 0.25 	imes 2 = 1.10$$

Question 4

A company tracks the number of complaints received daily, with the following probability distribution:

$$\begin{align*} X &: 0, 1, 2, 3 \ P(X) &: 0.45, 0.30, 0.20, 0.05 \end{align*}$$

What is the variance of $$X$$?

Accepted Answer

$$\sum (x - \mu)^2$ P(x)$$ where $$\mu = 0 	imes 0.45 + 1 	imes 0.30 + 2 	imes 0.20 + 3 	imes 0.05 = 0.85$$, 
ewline 	ext{Variance} = $$(0-0.85)^2$$ 	imes 0.45 + $$(1-0.85)^2$$ 	imes 0.30 + $$(2-0.85)^2$$ 	imes 0.20 + $$(3-0.85)^2$$ 	imes 0.05 = 0.7275$

Question 5

Which of the following scenarios describes a binomial experiment?

Accepted Answer

Counting the number of times a die lands on 2 in 10 rolls

Question 6

A pharmaceutical company claims the probability of gaining immunity from a new vaccine is 80%. The vaccine is given to 100 patients. What are the values of $$n$$ and $$p$$ for the binomial distribution?

Accepted Answer

$$n = 100, p = 0.80$$

Question 7

What is the probability of getting exactly 3 red marbles in 4 draws with replacement from a bag where the probability of red is 0.8?

Accepted Answer

$$\binom{4}{3} (0.8)^3$ (0.2)^1$ = 4 	imes 0.512 	imes 0.2 = 0.4096$$

Question 8

A company offers a new product, and 60% of customers are expected to purchase it after a demonstration. If you randomly select 10 customers, what is the expected number of purchases?

Accepted Answer

$$\mu = n p = 10 	imes 0.6 = 6$$

Question 9

Which formula gives the probability of exactly $$k$$ successes in $$n$$ binomial trials with probability $$p of $$success?

Accepted Answer

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

Question 10

A baker observes an average of 2 customers every 15 minutes. What is the probability that exactly 4 customers enter in a 15-minute period? (Use the Poisson distribution.)

Accepted Answer

$$P(X=4) = \frac{2^4$ e$$^{-2}$$}{4!} = \frac{16 e$$^{-2}$$}{24} \approx 0.0902$$

Question 11

A quality control inspector knows that, on average, there are 0.5 defects per meter of fabric. What is the probability that the inspector finds 0 defects in a given meter?

Accepted Answer

$$P(X=0) = \frac{0.5^0$ e$$^{-0.5}$$}{0!} = e$$^{-0.5}$$ \approx 0.6065$$

Question 12

Which of the following situations is best modeled by a hypergeometric distribution?

Accepted Answer

Drawing 5 cards from a deck of 52 without replacement and counting the number of aces

Question 13

A school has 386 tickets, one for each student. Tickets are removed from the pool after being chosen and 5 tickets are drawn. Which distribution should be used to model the number of winners from a particular class?

Accepted Answer

Hypergeometric

Question 14

A shipment contains 100 units, 5 of which are defective. If 20 units are randomly selected without replacement, what is the probability that exactly 2 are defective? (Use the hypergeometric formula.)

Accepted Answer

$$P(X=2) = \frac{\binom{5}{2} \binom{95}{18}}{\binom{100}{20}}$$

Question 15

A customer service call center receives an average of 12 calls per hour. What is the expected number of calls in a 30-minute period?

Accepted Answer

$$\lambda = 12 	imes 0.5 = 6$$

Question 16

Which of the following is NOT a requirement for using the Poisson distribution to approximate the binomial distribution?

Accepted Answer

$$n \geq 100$$ and $$np \leq 10$$

Question 17

A student working on a transportation project observes an average of 17.6 cars per minute at an intersection. What is the expected number of cars in a 20-minute period?

Accepted Answer

$$17.6 	imes 20 = 352$$

Discrete Probability Distributions: Study Guide

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