Q: A fair coin is flipped 5 times. What is the probability of getting exactly 3 heads?

$$\displaystyle \binom{5}{3} (0.5)^3$ (0.5)^2$ = 10 \times 0.125 \times 0.25 = 0.3125$$

Q: 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?

$$\displaystyle \binom{4}{2} (0.4)^2$ (0.6)^2$ = 6 \times 0.16 \times 0.36 = 0.3456$$

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 5 times. What is the probability of getting exactly 3 heads?

Accepted Answer

$$\displaystyle \binom{5}{3} (0.5)^3$ (0.5)^2$ = 10 	imes 0.125 	imes 0.25 = 0.3125$$

Question 4

Suppose $$X \sim \mathrm{Bin}(n, p). $$Which formula gives the mean of $$X$$?

Accepted Answer

$$np$$

Question 5

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

$$\displaystyle \binom{4}{2} (0.4)^2$ (0.6)^2$ = 6 	imes 0.16 	imes 0.36 = 0.3456$$

Question 6

Which of the following scenarios is NOT a binomial experiment?

Accepted Answer

Drawing 10 cards from a deck without replacement and counting the number of red cards.

Question 7

For a binomial random variable $$X \sim \mathrm{Bin}(n, p)$$, what is the standard deviation?

Accepted Answer

$$\sqrt{np(1-p)}$$

Question 8

If $$X is $$the number of successes in $$n$$ independent Bernoulli trials with probability $$p of $$success, what is the probability mass function (pmf) of $$X$$?

Accepted Answer

$$P(X = x) = \displaystyle \binom{n}{x} p^x (1-p$$)^{n-x}$$, for x = 0$$, 1, \ldots, n$$

Question 9

A researcher claims that more than 35% of car-owning households have three or more cars. In a sample of 400 households, 162 have three or more cars. Using the empirical rule, is this result unusual? (Assume $$p = 0.35)$$

Accepted Answer

Yes, because 162 is outside the interval $$[120.9, 159.1]$$ calculated using $$\mu = np$$ and $$\sigma = \sqrt{np(1-p)}.$$

Question 10

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

Accepted Answer

Left-skewed

Question 11

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 12

If $$X$$ and $$Y$$ are random variables such that $$Y = n - X (e.g$$., number of tails in $$n$$ coin tosses), what is $$\mathrm{Var}(X + Y)$$?

Accepted Answer

0

Question 13

A Bernoulli random variable $$X$$ has $$P(X=1) = 0.7. $$What is $$E[X]$$?

Accepted Answer

0.7

Question 14

Which of the following real-world scenarios can be modeled by a binomial distribution?

Accepted Answer

Counting the number of defective products in a batch of 50, where each product is tested independently and has the same probability of being defective.

Question 15

Suppose $$X \sim \mathrm{Bin}(n, p). If$$ $$p = 0.5$$, what is the shape of the distribution as $$n$$ increases?

Accepted Answer

It becomes more symmetric and bell-shaped.

Question 16

A factory tests 20 light bulbs and finds that 3 are defective. If the probability of a bulb being defective is 0.1, what is the expected number of defective bulbs in 20 tested bulbs?

Accepted Answer

$$20 	imes 0.1 = 2$$

Question 17

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

Accepted Answer

$$np(1-p)$$

Question 18

In a binomial experiment with $$n = 10$$ and $$p = 0.2$$, what is the probability of getting exactly 0 successes?

Accepted Answer

$$(1-0.2)^{10}$$ = $$0.8^{10}$$ \approx 0.1074$$

Question 19

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.

Lecture 19: Binomial Distribution – Study Notes

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