Q: Given the following probability distribution for the number of kids per household: 0 kids (0.15), 1 kid (0.60), 2 kids (0.25). What is the expected value (mean) number of kids per household?

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

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 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: 0 kids (0.15), 1 kid (0.60), 2 kids (0.25). What is the expected value (mean) number of kids per household?

Accepted Answer

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

Question 4

Which formula correctly calculates the variance $$\$$sigma^2$$ of a discrete random variable X$ with probability distribution P(x$$)$$?

Accepted Answer

$$\$$sigma^2$$ = \sum [(x - \mu)^2$ \cdot P(x)]$$

Question 5

A company tracks the number of complaints received daily: 0 (0.45), 1 (0.30), 2 (0.20), 3 (0.05). What is the standard deviation of the number of complaints?

Accepted Answer

$$\sqrt{\sum (x - \mu)^2$ P(x)}$$

Question 6

Which of the following scenarios is a binomial experiment?

Accepted Answer

Counting the number of heads in 10 coin flips

Question 7

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

Accepted Answer

$$n = 100, p = 0.80$$

Question 8

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

Accepted Answer

$$\binom{4}{3} (0.8)^3$ (0.2)^1$

Question 9

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

$$10 	imes 0.6 = 6$$

Question 10

Which formula gives the standard deviation $$\sigma of a $$binomial distribution with parameters $$n$$ and $$p$$?

Accepted Answer

$$\sigma = \sqrt{npq}$$

Question 11

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 formula.)

Accepted Answer

$$\frac{2^4$ e$$^{-2}$$}{4!}$$

Question 12

A quality control inspector knows there are on average 0.5 defects per meter of fabric. What is the probability that there are no defects in a meter? (Poisson distribution)

Accepted Answer

$$\frac{0.5^0$ e$$^{-0.5}$$}{0!}$$

Question 13

A school raffle draws 5 tickets from 386 students. If tickets are replaced after each draw, which distribution should be used to model the number of winners from a particular class?

Accepted Answer

Binomial distribution

Question 14

Which formula is used to calculate the probability of $$x$$ successes in a hypergeometric distribution?

Accepted Answer

$$\frac{\binom{r}{x} \binom{N-r}{n-x}}{\binom{N}{n}}$$

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

6

Question 16

Which of the following best describes when to use the Poisson distribution as an approximation to the binomial distribution?

Accepted Answer

When $$n is $$large and $$p is $$small

Question 17

A student analyzes traffic flow at an intersection for 20 minutes. If the average number of cars per minute is 17.6, what is the expected number of cars in 20 minutes?

Accepted Answer

$$17.6 	imes 20 = 352$$

Question 18

A quality control manager samples 20 units from a shipment of 100 units (5 defective) without replacement. Which distribution should be used to model the number of defective units found?

Accepted Answer

Hypergeometric distribution

Discrete Probability Distributions: Binomial, Poisson, and Hypergeometric Distributions

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