Q: In Pittsburgh, about 56% of the days in June are cloudy. For $$n = 30$$, $$p = 0.56$$, $$q = 0.44$$, what is the mean number of cloudy days?

$$\mu = 30 \times 0.56 = 16.8$$

Q: For the same Pittsburgh cloudy days scenario ($$n = 30$$, $$p = 0.56$$, $$q = 0.44$$), what is the standard deviation?

$$\sigma = \sqrt{30 \times 0.56 \times 0.44} \approx 2.7$$

Question 1

Which of the following is NOT a characteristic of a binomial experiment?

Accepted Answer

The probability of success changes from trial to trial.

Question 2

A doctor performs a surgical procedure with an 85% success rate on eight patients. Which values correctly represent n, p, and q for this binomial experiment?

Accepted Answer

$$n = 8$$, $$p = 0.85$$, $$q = 0.15$$

Question 3

Which scenario below is NOT a binomial experiment?

Accepted Answer

Selecting three marbles from a jar without replacement and counting the number of red marbles.

Question 4

What is the binomial probability formula for finding the probability of exactly $$x$$ successes in $$n$$ trials?

Accepted Answer

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

Question 5

A rotator cuff surgery has a 90% chance of success. The surgery is performed on three patients. What is the probability that exactly two surgeries are successful?

Accepted Answer

$$P(2) = \frac{3!}{2!1!} (0.9)^2$ (0.1)^1$

Question 6

In a survey, 48% of U.S. adults go online several times a day. If six adults are randomly selected, what is the probability that exactly four of them go online several times a day?

Accepted Answer

$$P(4) = \frac{6!}{4!2!} (0.48)^4$ (0.52)^2$

Question 7

A survey found that 75% of U.S. adults are experiencing digital device fatigue. You randomly select 100 adults. What is the probability that exactly 66 are experiencing digital device fatigue?

Accepted Answer

$$P(66) = \frac{100!}{66!34!} (0.75$$)^{66}$$ (0.25$$)^{34}$$

Question 8

A survey found that 22% of U.S. adults say that the economy, unemployment, and jobs are the most important problems. You randomly select four adults. What is the probability that at least two of them respond yes?

Accepted Answer

$$P(x \geq 2) = P(2) + P(3) + P(4)$$

Question 9

Which formula gives the mean of a binomial distribution?

Accepted Answer

$$\mu = n p$$

Question 10

Which formula gives the standard deviation of a binomial distribution?

Accepted Answer

$$\sigma = \sqrt{n p q}$$

Question 11

In Pittsburgh, about 56% of the days in June are cloudy. For $$n = 30$$, $$p = 0.56$$, $$q = 0.44$$, what is the mean number of cloudy days?

Accepted Answer

$$\mu = 30 	imes 0.56 = 16.8$$

Question 12

For the same Pittsburgh cloudy days scenario ($$n = 30$$, $$p = 0.56$$, $$q = 0.44$$), what is the standard deviation?

Accepted Answer

$$\sigma = \sqrt{30 	imes 0.56 	imes 0.44} \approx 2.7$$

Question 13

Which value would be considered unusual for the number of cloudy days in June in Pittsburgh, based on the mean and standard deviation?

Accepted Answer

11 cloudy days or fewer

Question 14

If the probability of success in a binomial experiment is $$p = 0.22$$ and the number of trials is $$n = 4$$, what is the probability of exactly two successes?

Accepted Answer

$$P(2) = \frac{4!}{2!2!} (0.22)^2$ (0.78)^2$

Question 15

Which of the following statements about the sum of probabilities in a binomial distribution is correct?

Accepted Answer

The sum of all probabilities for possible values of $$x is 1.$$

Question 16

Which technology tool is commonly used to calculate binomial probabilities, as shown in the study materials?

Accepted Answer

StatCrunch

Question 17

If a binomial experiment has $$n = 6$$, $$p = 0.64$$, and $$q = 0.36$$, what are the possible values for the random variable $$x$$?

Accepted Answer

$$x = 0, 1, 2, 3, 4, 5, 6$$

Discrete Probability Distributions: Binomial Experiments and Binomial Probability

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