Q: A pharmaceutical company tests a new drug on 8 patients. Each patient has a 0.25 probability of experiencing a side effect. What is the probability that exactly 3 patients experience the side effect? Use the binomial probability formula.

$$P(3) = \frac{8!}{3!5!} (0.25)^3$ (0.75)^5$ = 56 \times 0.015625 \times 0.2373046875 = 0.208$$

Q: In a binomial experiment, if $$n = 12$$, $$p = 0.4$$, and $$q = 0.6$$, what is the mean number of successes?

$$\mu = np = 12 \times 0.4 = 4.8$$

Q: A survey finds that 5% of adults are vegetarians. If 5 adults are randomly selected, what is the probability that exactly 2 are vegetarians?

$$P(2) = \frac{5!}{2!3!} (0.05)^2$ (0.95)^3$ = 10 \times 0.0025 \times 0.857375 = 0.021$$

Q: A digital channel transmits 5 bits, each with a 10% chance of error. What is the probability that at least one bit is received in error?

$$P(\text{at least one error}) = 1 - P(0) = 1 - (0.9)^5$ = 1 - 0.59049 = 0.40951$$

Q: If a binomial experiment has $$n = 20$$ trials and the probability of success $$p = 0.3$$, what is the variance?

$$\$$sigma^2$$ = npq = 20 \times 0.3 \times 0.7 = 4.2$$

Q: A fair coin is flipped 10 times. What is the probability of getting exactly 4 heads?

$$P(4) = \frac{10!}{4!6!} (0.5)^4$ (0.5)^6$ = 210 \times 0.0625 \times 0.015625 = 0.205$$

Q: A rare molecule is found in 3% of air samples. If 20 samples are analyzed, what is the expected number of samples containing the molecule?

$$\mu = np = 20 \times 0.03 = 0.6$$

Question 1

Which of the following scenarios best fits the requirements for a binomial probability distribution?

Accepted Answer

Testing 15 light bulbs to see how many fail within a year, assuming each bulb has the same probability of failure and the outcomes are independent.

Question 2

Which of the following is NOT a requirement for a binomial probability distribution?

Accepted Answer

The number of trials must be unlimited.

Question 3

A pharmaceutical company tests a new drug on 8 patients. Each patient has a 0.25 probability of experiencing a side effect. What is the probability that exactly 3 patients experience the side effect? Use the binomial probability formula.

Accepted Answer

$$P(3) = \frac{8!}{3!5!} (0.25)^3$ (0.75)^5$ = 56 	imes 0.015625 	imes 0.2373046875 = 0.208$$

Question 4

In a binomial experiment, if $$n = 12$$, $$p = 0.4$$, and $$q = 0.6$$, what is the mean number of successes?

Accepted Answer

$$\mu = np = 12 	imes 0.4 = 4.8$$

Question 5

A survey finds that 5% of adults are vegetarians. If 5 adults are randomly selected, what is the probability that exactly 2 are vegetarians?

Accepted Answer

$$P(2) = \frac{5!}{2!3!} (0.05)^2$ (0.95)^3$ = 10 	imes 0.0025 	imes 0.857375 = 0.021$$

Question 6

Which formula gives the standard deviation for a binomial distribution?

Accepted Answer

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

Question 7

A digital channel transmits 5 bits, each with a 10% chance of error. What is the probability that at least one bit is received in error?

Accepted Answer

$$P(	ext{at least one error}) = 1 - P(0) = 1 - (0.9)^5$ = 1 - 0.59049 = 0.40951$$

Question 8

If a binomial experiment has $$n = 20$$ trials and the probability of success $$p = 0.3$$, what is the variance?

Accepted Answer

$$\$$sigma^2$$ = npq = 20 	imes 0.3 	imes 0.7 = 4.2$$

Question 9

A fair coin is flipped 10 times. What is the probability of getting exactly 4 heads?

Accepted Answer

$$P(4) = \frac{10!}{4!6!} (0.5)^4$ (0.5)^6$ = 210 	imes 0.0625 	imes 0.015625 = 0.205$$

Question 10

In a binomial experiment, what does the term 'success' refer to?

Accepted Answer

Any outcome that is considered favorable or of interest, regardless of whether it is 'good' or 'bad'.

Question 11

A rare molecule is found in 3% of air samples. If 20 samples are analyzed, what is the expected number of samples containing the molecule?

Accepted Answer

$$\mu = np = 20 	imes 0.03 = 0.6$$

Question 12

Using the range rule of thumb, which value is considered significantly high for a binomial distribution with $$\mu = 230$$ and $$\sigma = 10.7$$?

Accepted Answer

$$\mu + 2\sigma = 230 + 2 	imes 10.7 = 251.4$$

Question 13

In a group of 460 NFL overtime games, the mean number of wins is 230 and the standard deviation is 10.7. Is 252 wins considered significantly high?

Accepted Answer

Yes, because 252 \u003e $$\mu + 2\sigma = 251.4$$

Question 14

If the probability of a randomly selected adult smartphone owner being cashless is 0.05, what is the probability that none out of 10 randomly selected adults are cashless?

Accepted Answer

$$P(0) = (0.95$$)^{10}$$ = 0.5987$$

Question 15

Which of the following best describes the independence requirement in a binomial experiment?

Accepted Answer

The outcome of any individual trial does not affect the probabilities in the other trials.

Question 16

A binomial experiment is conducted with $$n = 7$$ and $$p = 0.6. $$What is the probability of getting exactly 5 successes?

Accepted Answer

$$P(5) = \frac{7!}{5!2!} (0.6)^5$ (0.4)^2$ = 21 	imes 0.07776 	imes 0.16 = 0.261$$

Question 17

If a binomial experiment has $$n = 15$$ and $$p = 0.2$$, what is the probability of getting fewer than 3 successes?

Accepted Answer

$$P(0) + P(1) + P(2)$$, where $$P(x) = \frac{15!}{x!(15-x)!} (0.2)^x (0.8$$)^{15-x}$$

Binomial Probability Distributions – 5.2:Essentials of Statistics (Chapter 5 Study Notes)

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