Q: A factory tests 12 products, each with a 0.9 probability of passing quality control. What is the expected number of products that pass?

$$\mu = np = 12 \times 0.9 = 10.8$$

Q: If $$n=20$$ and $$p=0.4$$, what is the standard deviation of the number of successes in a binomial experiment?

$$\sigma = \sqrt{np(1-p)} = \sqrt{20 \times 0.4 \times 0.6} \approx 2.19$$

Question 1

Which of the following scenarios is most appropriately modeled by a binomial distribution?

Accepted Answer

Counting the number of heads in 20 coin tosses.

Question 2

Which of the following is NOT a required condition for using the binomial distribution?

Accepted Answer

The trials must be dependent on each other.

Question 3

Given a binomial experiment with $$n=8$$ trials and probability of success $$p=0.3$$, what is the probability of exactly 2 successes? Use the binomial formula.

Accepted Answer

$$P(2) = \frac{8!}{2!6!} (0.3)^2$ (0.7)^6$ \approx 0.296$$

Question 4

A factory tests 12 products, each with a 0.9 probability of passing quality control. What is the expected number of products that pass?

Accepted Answer

$$\mu = np = 12 	imes 0.9 = 10.8$$

Question 5

If $$n=20$$ and $$p=0.4$$, what is the standard deviation of the number of successes in a binomial experiment?

Accepted Answer

$$\sigma = \sqrt{np(1-p)} = \sqrt{20 	imes 0.4 	imes 0.6} \approx 2.19$$

Question 6

A survey finds that 60% of people prefer tea over coffee. If 5 people are randomly selected, what is the probability that exactly 3 prefer tea?

Accepted Answer

$$P(3) = \frac{5!}{3!2!} (0.6)^3$ (0.4)^2$ \approx 0.3456$$

Question 7

In a binomial experiment, what does the term $$n!$$ represent in the probability formula?

Accepted Answer

The number of ways to arrange $$n$$ items.

Question 8

A medical test has a 95% chance of correctly identifying a disease. If 4 patients are tested, what is the probability that all 4 tests are correct?

Accepted Answer

$$P(4) = (0.95)^4$ \approx 0.8145$$

Question 9

Which of the following is a real-world example where the binomial distribution would NOT be appropriate?

Accepted Answer

Measuring the weight of apples in a harvest.

Question 10

A group claims that female employees are less likely to be promoted. In a random sample of 10 promotions from a company with 50% female employees, none are female. What is the probability of this outcome under random selection?

Accepted Answer

$$P(0) = \frac{10!}{0!10!} (0.5)^0$ (0.5$$)^{10}$$ = 0.0009766$$

Question 11

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

Accepted Answer

$$P(x) = \frac{n!}{x!(n-x)!} p^x (1-p$$)^{n-x}$$

Question 12

In the NYPD racial profiling example, if the mean number of non-white confrontations is 277,000 and the standard deviation is 351, what range would you expect nearly all values to fall within according to the empirical rule?

Accepted Answer

$$277,000 \pm 3 	imes 351 = [275,947, 278,053]$$

Question 13

If a binomial random variable $$X$$ has $$n=15$$ and $$p=0.2$$, what is the mean and standard deviation?

Accepted Answer

$$\mu = 15 	imes 0.2 = 3$$, $$\sigma = \sqrt{15 	imes 0.2 	imes 0.8} \approx 1.55$$

Question 14

Which of the following statements about binomial trials is TRUE?

Accepted Answer

The outcome of one trial does not affect the outcome of another trial.

Question 15

A student flips a coin 6 times. What is the probability of getting exactly 4 heads?

Accepted Answer

$$P(4) = \frac{6!}{4!2!} (0.5)^4$ (0.5)^2$ = 15 	imes 0.0625 	imes 0.25 = 0.234375$$

Question 16

Which of the following is the correct interpretation of $$p in $$the binomial distribution?

Accepted Answer

The probability of success on a single trial.

Question 17

If you want to use the binomial distribution to model the number of people who vote 'yes' in a referendum, which condition must be met?

Accepted Answer

Each person's vote must be independent of others.

Question 18

A bag contains 10 marbles, 4 red and 6 blue. If you randomly select one marble and record whether it is red, which binomial parameter does the probability of selecting a red marble represent?

Accepted Answer

$$p = 0.4$$

(Lecture 15) Binomial Distribution and Binary Outcomes in Statistics

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