Q: For a geometric probability model, which formula gives the probability that the first success occurs on the $$xth $$trial?

$$P(X = x) = p \times q$$^{x-1}$$

Q: If $$p = 0.3$$ and $$q = 0.7$$, what is the standard deviation ($$\sigma) of a $$binomial distribution with $$n = 50$$ trials?

$$\sigma = \sqrt{50 \times 0.3 \times 0.7}$$

Q: A student rolls a die repeatedly until a '6' appears. What is the probability that the first '6' occurs on the 4th roll?

$$P(X = 4) = (\frac{5}{6}$$)^{3}$$ \times \frac{1}{6}$$

Q: A factory produces light bulbs with a 2% defect rate. What is the probability that the first defective bulb is found on the 5th bulb tested?

$$P(X = 5) = 0.02 \times 0.98$$^{4}$$

Question 1

Which of the following best describes a Bernoulli trial?

Accepted Answer

A trial where each observation has exactly two possible outcomes, and the probability of success is the same for each trial.

Question 2

What is the 10% Condition in the context of sampling without replacement?

Accepted Answer

It is required that the sample size be less than 10% of the population to ensure independence.

Question 3

A researcher flips a coin 20 times and records the number of heads. Which probability model is most appropriate for this scenario?

Accepted Answer

Binomial model

Question 4

Which formula gives the mean ($$\mu) of a $$binomial distribution with $$n$$ trials and probability of success $$p$$?

Accepted Answer

$$\mu = n 	imes p$$

Question 5

For a geometric probability model, which formula gives the probability that the first success occurs on the $$xth $$trial?

Accepted Answer

$$P(X = x) = p 	imes q$$^{x-1}$$

Question 6

A medical test for a rare disease returns positive in 2 out of 1000 healthy people. If the test is performed on 5000 healthy people, what is the expected number of false positives?

Accepted Answer

10

Question 7

Which of the following is NOT a requirement for using the binomial model?

Accepted Answer

Each trial must have more than two outcomes.

Question 8

If $$p = 0.3$$ and $$q = 0.7$$, what is the standard deviation ($$\sigma) of a $$binomial distribution with $$n = 50$$ trials?

Accepted Answer

$$\sigma = \sqrt{50 	imes 0.3 	imes 0.7}$$

Question 9

A student rolls a die repeatedly until a '6' appears. What is the probability that the first '6' occurs on the 4th roll?

Accepted Answer

$$P(X = 4) = (\frac{5}{6}$$)^{3}$$ 	imes \frac{1}{6}$$

Question 10

Which condition must be met to use the Normal model as a good approximation for a Binomial model?

Accepted Answer

Both $$np \geq 10$$ and $$nq \geq 10$$

Question 11

A scientist claims a result is statistically significant. What does this mean?

Accepted Answer

The result has a very low probability of occurring by chance.

Question 12

In a binomial experiment with $$n = 20$$ and $$p = 0.4$$, what is the expected number of successes?

Accepted Answer

8

Question 13

Which of the following scenarios is best modeled by a geometric probability model?

Accepted Answer

Counting the number of trials until the first success occurs.

Question 14

A factory produces light bulbs with a 2% defect rate. What is the probability that the first defective bulb is found on the 5th bulb tested?

Accepted Answer

$$P(X = 5) = 0.02 	imes 0.98$$^{4}$$

Question 15

Which of the following is a key difference between the binomial and geometric models?

Accepted Answer

The binomial model counts the number of successes in a fixed number of trials, while the geometric model counts the number of trials until the first success.

Question 16

If a binomial experiment has $$n = 100$$, $$p = 0.05$$, and $$q = 0.95$$, what is the standard deviation?

Accepted Answer

$$\sigma = \sqrt{100 	imes 0.05 	imes 0.95}$$

Question 17

A quality control inspector checks batteries and finds that the probability a battery is defective is 0.01. What is the probability that the first defective battery is found on the 8th battery checked?

Accepted Answer

$$P(X = 8) = 0.01 	imes 0.99$$^{7}$$

Question 18

Which of the following statements about statistical significance is correct?

Accepted Answer

A statistically significant result is unlikely to have occurred by random chance.

Question 19

A binomial experiment is conducted with $$n = 30$$ and $$p = 0.2. $$What is the mean and standard deviation?

Accepted Answer

Mean: $$6$$, Standard deviation: $$\sqrt{30 	imes 0.2 	imes 0.8}$$

Bernoulli Trials, Binomial and Geometric Probability Models

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