find the indicated probabilities using the geometric distribution, the poisson distribution, or the binomial distribution. then determine if the events are unusual. if convenient, use the appropriate probability table or technology to find the probabilities.

fifty-one percent of u.s. adults oppose hydraulic fracturing (fracking) as a means of increasing the production of natural gas and oil in the united states. you randomly select five u.s. adults. find the probability that the number of u.s. adults who oppose fracking as a means of increasing the production of natural gas and oil in the united states is (a) exactly two, (b) less than four, and (c) at least three.

(a) p(2) = 0.306
(round to three decimal places as needed.)

(b) p(less than four) = 0.8
(round to three decimal places as needed.)

(c) p(at least three) = 0.519
(round to three decimal places as needed.)

which of the events are unusual? select all that apply.

a. the event p(2) is unusual.
b. the event p(less than four) is unusual.
c. the event p(at least three) is unusual.
d. none of the events are unusual.

Question

find the indicated probabilities using the geometric distribution, the poisson distribution, or the binomial distribution. then determine if the events are unusual. if convenient, use the appropriate probability table or technology to find the probabilities.

fifty-one percent of u.s. adults oppose hydraulic fracturing (fracking) as a means of increasing the production of natural gas and oil in the united states. you randomly select five u.s. adults. find the probability that the number of u.s. adults who oppose fracking as a means of increasing the production of natural gas and oil in the united states is (a) exactly two, (b) less than four, and (c) at least three.

(a) p(2) = 0.306
(round to three decimal places as needed.)

(b) p(less than four) = 0.8
(round to three decimal places as needed.)

(c) p(at least three) = 0.519
(round to three decimal places as needed.)

which of the events are unusual? select all that apply.

a. the event p(2) is unusual.
b. the event p(less than four) is unusual.
c. the event p(at least three) is unusual.
d. none of the events are unusual.

Accepted Answer

{ "quality": "clear", "question_count": 4, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Binomial Probability Formula", "Cumulative Binomial Probability", "Binomial Distribution Unusual Values" ], "current_concepts": [ "Binomial Probability Formula", "Cumulative Binomial Probability", "Binomial Distribution Unusual Values" ] } ## Identify binomial parameters and probability formula $$ \begin{aligned} &n = 5,\quad p = 0.51,\quad q = 1 - p = 0.49\ &P(x) = \binom{5}{x} (0.51)^x (0.49)^{5-x} \end{aligned} $$ ## Calculate probabilities for parts (a), (b), and (c) $$ \begin{aligned} &P(0) = \binom{5}{0}(0.51)^0(0.49)^5 \approx 0.0282\ &P(1) = \binom{5}{1}(0.51)^1(0.49)^4 \approx 0.1470\ &P(2) = \binom{5}{2}(0.51)^2(0.49)^3 \approx 0.3060 \approx 0.306\ &P(3) = \binom{5}{3}(0.51)^3(0.49)^2 \approx 0.3185\ &P(4) = \binom{5}{4}(0.51)^4(0.49)^1 \approx 0.1657\ &P(5) = \binom{5}{5}(0.51)^5(0.49)^0 \approx 0.0345\ &P(\text{less than four}) = P(0) + P(1) + P(2) + P(3) \approx 0.0282 + 0.1470 + 0.3060 + 0.3185 = 0.7997 \approx 0.800\ &P(\text{at least three}) = P(3) + P(4) + P(5) \approx 0.3185 + 0.1657 + 0.0345 = 0.5187 \approx 0.519 \end{aligned} $$ ## Determine unusual events $$ \begin{aligned} &\text{An event is unusual if its probability is less than or equal to } 0.05.\ &P(2) = 0.306 > 0.05\ &P(\text{less than four}) = 0.800 > 0.05\ &P(\text{at least three}) = 0.519 > 0.05\ &\text{None of the events are unusual.} \end{aligned} $$ ## Question 1 (a) $P(2) =$ 0.306 ## Question 2 (b) $P(\text{less than four}) =$ 0.800 ## Question 3 (c) $P(\text{at least three}) =$ 0.519 ## Question 4 A. The event P(2) is unusual.

B. The event P(less than four) is unusual.

C. The event P(at least three) is unusual.

D. None of the events are unusual.

{ "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Statistics", "Binomial Probability Formula" ] }

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QUESTION IMAGE

Question

Explanation:

Identify binomial parameters and probability formula

Calculate probabilities for parts (a), (b), and (c)

Determine unusual events

Answer:

Question 1

Question 2

Question 3

Question 4