Q: A binomial experiment is performed with $$n=10$$ and $$p=0.2. $$What is the standard deviation of the number of successes?

$$\sqrt{10 \times 0.2 \times 0.8}$$

Question 1

Which of the following is NOT a requirement for a binomial experiment?

Accepted Answer

Each trial must have more than two possible outcomes.

Question 2

Let $$X be a $$Bernoulli random variable with parameter $$p. $$What is the variance of $$X$$?

Accepted Answer

$$p(1-p)$$

Question 3

A fair coin is flipped 8 times. What is the probability of getting exactly 3 heads? (Use $$P(X = x) = {n \choose x} p^x (1-p$$)^{n-x}$$, with p=0.5$)

Accepted Answer

$$\displaystyle {8 \choose 3} (0.5)^3$ (0.5)^5$

Question 4

Which of the following scenarios is a binomial experiment?

Accepted Answer

Counting the number of defective bulbs in a sample of 20, where each bulb is tested and replaced before the next is tested.

Question 5

Suppose $$X \sim \mathrm{Bin}(n, p). $$What is the mean and standard deviation of $$X$$?

Accepted Answer

Mean: $$np$$, Standard deviation: $$\sqrt{np(1-p)}$$

Question 6

A population of pets is 40% cats and 60% dogs. If you randomly select a pet, record, and replace, repeating this 4 times, what is the probability of getting exactly 2 cats?

Accepted Answer

$$\displaystyle {4 \choose 2} (0.4)^2$ (0.6)^2$

Question 7

If $$X is a $$binomial random variable with $$n=20$$ and $$p=0.35$$, what is the expected number of successes?

Accepted Answer

$$7$$

Question 8

Which of the following best describes the shape of a binomial distribution with $$n=8$$ and $$p=0.85$$?

Accepted Answer

Left-skewed

Question 9

For a binomial distribution $$X \sim \mathrm{Bin}(n, p)$$, under what condition will the distribution be approximately bell-shaped?

Accepted Answer

$$np(1-p) \geq 10$$

Question 10

A researcher samples 400 car-owning households and finds 162 have three or more cars. If the true proportion is $$p=0.35$$, is this result unusual? (Use the Empirical Rule and $$\mu = np$$, $$\sigma = \sqrt{np(1-p)})$$

Accepted Answer

Yes, because 162 is outside the interval $$[120.9, 159.1] (i.e$$., more than 2 standard deviations from the mean).

Question 11

If $$X$$ and $$Y$$ are random variables such that $$X is $$the number of heads and $$Y is $$the number of tails in 4 tosses of a fair coin, what is $$\mathrm{Var}(X+Y)$$?

Accepted Answer

$$0$$

Question 12

Which of the following is the probability mass function (pmf) for a Bernoulli random variable $$X$$ with parameter $$p$$?

Accepted Answer

$$f(x) = p^x (1-p$$)^{1-x}$$, x=0$$,1$$

Question 13

A binomial experiment is performed with $$n=10$$ and $$p=0.2. $$What is the standard deviation of the number of successes?

Accepted Answer

$$\sqrt{10 	imes 0.2 	imes 0.8}$$

Question 14

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

Accepted Answer

As $$n$$ increases and $$p is $$not too close to 0 or 1, the binomial distribution becomes approximately bell-shaped.

Question 15

Suppose you flip a fair coin 5 times. What is the probability of getting at least 1 head?

Accepted Answer

$$1 - (0.5)^5$

Question 16

If $$X \sim \mathrm{Bin}(n, p)$$, which of the following is the correct formula for the probability of exactly $$x$$ successes?

Accepted Answer

$$P(X = x) = {n \choose x} p^x (1-p$$)^{n-x}$$

Question 17

A quality control inspector tests 15 products, each with a 10% chance of being defective. What is the probability that exactly 2 are defective?

Accepted Answer

$$\displaystyle {15 \choose 2} (0.1)^2$ (0.9$$)^{13}$$

Question 18

Which of the following best explains why the mean of a binomial random variable $$X \sim \mathrm{Bin}(n, p) is$$ $$np$$?

Accepted Answer

Because a binomial random variable is the sum of $$n$$ independent Bernoulli random variables, each with mean $$p.$$

Question 19

If a binomial distribution has $$n=12$$ and $$p=0.5$$, what is the probability of getting exactly 6 successes?

Accepted Answer

$$\displaystyle {12 \choose 6} (0.5$$)^{12}$$

Binomial and Bernoulli Distributions: Definitions, Properties, and Applications

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