Q: If $$p = 0.8$$ and $$n = 10$$, what is the mean number of successes in a binomial experiment?

$$\mu = np = 10 \times 0.8 = 8$$

Q: Given $$n = 200$$ and $$p = 0.83$$, what is the standard deviation of the number of successes in a binomial experiment?

$$\sigma = \sqrt{npq} = \sqrt{200 \times 0.83 \times 0.17} \approx 5.31$$

Q: A binomial experiment consists of 16 trials with $$p = 0.45. $$What is the variance of the number of successes?

$$\$$sigma^2$$ = npq = 16 \times 0.45 \times 0.55 = 3.96$$

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

A researcher flips a coin 10 times and records the number of heads. Which parameter represents the probability of getting heads in a single flip?

Accepted Answer

$$p$$

Question 3

In a binomial experiment, what does the variable $$x$$ represent?

Accepted Answer

The number of successes in $$n$$ trials.

Question 4

Which formula gives the probability of exactly $$x$$ successes in $$n$$ binomial trials?

Accepted Answer

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

Question 5

A burglar alarm system has 4 fail-safe components. The probability of each failing is 0.05. What is the probability that at most 2 will fail?

Accepted Answer

$$P(x \leq 2) = 0.998$$

Question 6

If $$p = 0.8$$ and $$n = 10$$, what is the mean number of successes in a binomial experiment?

Accepted Answer

$$\mu = np = 10 	imes 0.8 = 8$$

Question 7

A student guesses on a 15-question True/False exam. What is the probability of passing by getting at least 11 correct answers? (Assume $$p = 0.5)$$

Accepted Answer

$$P(x \geq 11) = 0.059$$

Question 8

Which of the following scenarios is NOT a binomial experiment?

Accepted Answer

Measuring the heights of 10 randomly selected students.

Question 9

Given $$n = 200$$ and $$p = 0.83$$, what is the standard deviation of the number of successes in a binomial experiment?

Accepted Answer

$$\sigma = \sqrt{npq} = \sqrt{200 	imes 0.83 	imes 0.17} \approx 5.31$$

Question 10

If the probability of success in a single trial is $$p$$, what is the probability of failure?

Accepted Answer

$$q = 1 - p$$

Question 11

A binomial experiment consists of 16 trials with $$p = 0.45. $$What is the variance of the number of successes?

Accepted Answer

$$\$$sigma^2$$ = npq = 16 	imes 0.45 	imes 0.55 = 3.96$$

Question 12

Which of the following best describes the meaning of 'success' in a binomial experiment?

Accepted Answer

It is an arbitrary label defined by the researcher for one of the two possible outcomes.

Question 13

A multiple-choice quiz contains 10 questions, each with 4 options. If a student guesses on all questions, is this a binomial experiment?

Accepted Answer

No, because each question has more than two possible outcomes.

Question 14

If $$n = 10$$, $$p = 0.8$$, and you want the probability of at least 9 successes, which calculation is correct?

Accepted Answer

$$P(x \geq 9) = P(x = 9) + P(x = 10)$$

Question 15

Which calculator function is used to compute $$\binom{n}{x} ($$the number of combinations) for binomial probabilities?

Accepted Answer

The $$nCr$$ function

Question 16

A medical test has an 85% chance of success. If 6 patients are tested, what is the probability that exactly 4 will succeed? (Use $$p = 0.85$$, $$n = 6$$, $$x = 4)$$

Accepted Answer

$$P(4) = \binom{6}{4} \cdot 0.85^4$ \cdot 0.15^2$

Question 17

Which of the following is the correct formula for the standard deviation of a binomial distribution?

Accepted Answer

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

Question 18

If you want to find the probability of 'at most' $$k$$ successes in a binomial experiment, which approach should you use?

Accepted Answer

Sum the probabilities for $$x = 0 up to$$ $$x = k.$$

Question 19

A binomial distribution table is most useful for which type of calculation?

Accepted Answer

Finding cumulative probabilities for a range of successes.

4.2- Binomial Distributions: Concepts, Formulas, and Applications

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