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 - two 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 seven 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 three, (b) less than four, and (c) at least three.
(a) ( p(3)=square )
(round to three decimal places as needed.)

Explanation:

Step1: Identify Distribution Type

This is a binomial distribution problem. Let \( n = 7 \) (number of trials), \( p = 0.52 \) (probability of success, opposing fracking), and we want \( P(X = 3) \). The binomial probability formula is \( P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \).

Step2: Calculate Binomial Coefficient

Calculate \( \binom{7}{3} = \frac{7!}{3!(7 - 3)!} = \frac{7!}{3!4!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \).

Step3: Compute Probability Terms

\( p^k = 0.52^3 \approx 0.140608 \), \( (1 - p)^{n - k} = 0.48^4 \approx 0.05308416 \).

Step4: Multiply Terms

\( P(3) = 35 \times 0.140608 \times 0.05308416 \approx 35 \times 0.007465 \approx 0.261275 \).

Answer:

\( 0.261 \) (rounded to three decimal places)