Q: For a binomial distribution with $$n = 30$$ and $$p = 0.62$$, what is the standard deviation?

$$\sigma = \sqrt{npq} = \sqrt{30 \times 0.62 \times 0.38}$$

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

$$1.10$$

Question 4

A factory produces lightbulbs in batches of 50. The probability distribution for the number of defective lightbulbs in a randomly selected batch is shown below:

$$\begin{align*} X &: 0, 1, 2, 3, 4, 5 \ P(X) &: 0.20, 0.30, 0.25, 0.15, 0.07, 0.03 \end{align*}$$

What is the expected number of defective bulbs per batch?

Accepted Answer

$$1.53$$

Question 5

Which of the following scenarios describes a binomial experiment?

Accepted Answer

Counting the number of heads in 10 flips of a fair coin

Question 6

A basketball player has a 70% chance of making a free throw. The player shoots until finally making a basket, where $$X is $$the number of shots taken. Is this a binomial experiment?

Accepted Answer

No, there are not a fixed number of trials

Question 7

You take a 6-question True/False quiz. What is the probability of getting all 6 questions correct by guessing?

Accepted Answer

$$\displaystyle \left(\frac{1}{2}\$$right)^6$$

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

$$6$$

Question 9

For a binomial distribution with $$n = 30$$ and $$p = 0.62$$, what is the standard deviation?

Accepted Answer

$$\sigma = \sqrt{npq} = \sqrt{30 	imes 0.62 	imes 0.38}$$

Question 10

A gardener plants 8 seeds, each with a 65% probability of germinating. What is the probability that less than 4 seeds germinate successfully? (Select the correct approach)

Accepted Answer

$$P(X \u003c 4) = \sum_{x=0}^{3} {8 \choose x} (0.65)^x (0.35$$)^{8-x}$$

Question 11

Which of the following is a scenario where the Poisson distribution is appropriate?

Accepted Answer

Counting the number of cars passing through an intersection in 10 minutes

Question 12

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

Accepted Answer

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

Question 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

Hypergeometric

Question 15

Which formula gives the probability of exactly $$x$$ successes in a hypergeometric distribution?

Accepted Answer

$$P(X=x) = \frac{\left( r \ x ight) \left( N-r \ n-x ight)}{\left( N \ n ight)}$$

Question 16

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

$$0.66$$

Question 17

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 18

In a large population of 10,000 lab mice, each mouse has an independent 0.0003 probability of carrying a rare genetic mutation. Can the number of mice with the mutation be approximated using the Poisson distribution, and if so, what is $$\lambda$$?

Accepted Answer

Yes, $$\lambda = 10,000 	imes 0.0003 = 3$$

Discrete Probability Distributions: Discrete, Binomial, Poisson, and Hypergeometric Distributions

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