Q: A fair coin is tossed 5 times. What is the probability of getting exactly 3 heads? Use the binomial probability formula.

$$P(X=3) = \frac{5!}{3!2!} \times (0.5)^3$ \times (0.5)^2$ = 0.3125$$

Q: A multiple-choice test has 10 questions, each with 4 possible answers, only one of which is correct. If a student guesses on all questions, what is the probability that they get exactly 2 questions correct?

$$P(X=2) = \frac{10!}{2!8!} \times (0.25)^2$ \times (0.75)^8$ \approx 0.2816$$

Q: A die is rolled 8 times. What is the probability of getting exactly 2 sixes? (Assume each roll is independent.)

$$P(X=2) = \frac{8!}{2!6!} \times (\frac{1}{6})^2$ \times (\frac{5}{6})^6$ \approx 0.2611$$

Q: A factory produces light bulbs with a 2% defect rate. If 50 bulbs are tested, what is the expected number of defective bulbs?

$$E(X) = np = 50 \times 0.02 = 1$$

Q: A sample of test scores is normally distributed with $$\mu = 80$$ and $$\sigma = 5. $$What score corresponds to the 97.5th percentile? (Use $$z = 1.96$$ for the 97.5th percentile.)

$$x = \mu + z\sigma = 80 + 1.96 \times 5 = 89.8$$

Question 1

A fair coin is tossed 5 times. What is the probability of getting exactly 3 heads? Use the binomial probability formula.

Accepted Answer

$$P(X=3) = \frac{5!}{3!2!} 	imes (0.5)^3$ 	imes (0.5)^2$ = 0.3125$$

Question 2

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

Accepted Answer

$$\mu = np$$

Question 3

A multiple-choice test has 10 questions, each with 4 possible answers, only one of which is correct. If a student guesses on all questions, what is the probability that they get exactly 2 questions correct?

Accepted Answer

$$P(X=2) = \frac{10!}{2!8!} 	imes (0.25)^2$ 	imes (0.75)^8$ \approx 0.2816$$

Question 4

Which of the following is the formula for the variance of a binomial distribution?

Accepted Answer

$$\$$sigma^2$$ = np(1-p)$$

Question 5

A die is rolled 8 times. What is the probability of getting exactly 2 sixes? (Assume each roll is independent.)

Accepted Answer

$$P(X=2) = \frac{8!}{2!6!} 	imes (\frac{1}{6})^2$ 	imes (\frac{5}{6})^6$ \approx 0.2611$$

Question 6

Which of the following is NOT an assumption of the binomial distribution?

Accepted Answer

The probability of success changes from trial to trial.

Question 7

A factory produces light bulbs with a 2% defect rate. If 50 bulbs are tested, what is the expected number of defective bulbs?

Accepted Answer

$$E(X) = np = 50 	imes 0.02 = 1$$

Question 8

Which formula would you use to find the probability of getting at most 3 successes in 7 independent Bernoulli trials with probability of success $$p$$?

Accepted Answer

$$P(X \leq 3) = \sum_{k=0}^{3} \frac{7!}{k!(7-k)!} p^k (1-p$$)^{7-k}$$

Question 9

A normal distribution has mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What is the $$z-$$score for a value of $$x = 130$$?

Accepted Answer

$$z = \frac{130 - 100}{15} = 2$$

Question 10

Which of the following statements about the standard normal distribution is TRUE?

Accepted Answer

It has a mean of 0 and a standard deviation of 1.

Question 11

If $$X is $$normally distributed with $$\mu = 50$$ and $$\sigma = 10$$, what is the probability that $$X is $$less than 40? (Use the standard normal table for $$z = -1.)$$

Accepted Answer

$$P(X \u003c 40) = P(z \u003c -1) \approx 0.1587$$

Question 12

Which formula is used to convert a raw score $$x to a$$ $$z-$$score in a normal distribution?

Accepted Answer

$$z = \frac{x - \mu}{\sigma}$$

Question 13

A sample of test scores is normally distributed with $$\mu = 80$$ and $$\sigma = 5. $$What score corresponds to the 97.5th percentile? (Use $$z = 1.96$$ for the 97.5th percentile.)

Accepted Answer

$$x = \mu + z\sigma = 80 + 1.96 	imes 5 = 89.8$$

Question 14

Which of the following is a property of the normal distribution?

Accepted Answer

It is symmetric about the mean.

Question 15

A population is normally distributed with $$\mu = 60$$ and $$\sigma = 8. $$What is the probability that a randomly selected value is greater than 76? (Use $$z = 2.)$$

Accepted Answer

$$P(X \u003e 76) = P(z \u003e 2) = 1 - 0.9772 = 0.0228$$

Question 16

Which formula is used to find the probability of a value between $$a$$ and $$b in a $$normal distribution?

Accepted Answer

$$P(a \u003c X \u003c b) = P(z_b) - P(z_a)$$

Question 17

A student scores 1.5 standard deviations above the mean on a standardized test. What percentage of students scored lower than this student? (Use the standard normal table for $$z = 1.5.)$$

Accepted Answer

$$P(z \u003c 1.5) \approx 0.9332$$

Question 18

Which of the following is the correct formula for the probability mass function of a binomial random variable $$X$$?

Accepted Answer

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

Discrete and Normal Probability Distributions: Study Notes

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