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 - six 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 six 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)=square )
(round to three decimal places as needed.)

Explanation:

Step1: Identify the distribution

This is a binomial probability problem. Let \( n = 6 \) (number of trials), \( p = 0.56 \) (probability of success, i.e., an adult opposes fracking), and we want \( P(X = 2) \). The binomial probability formula is \( P(X = k)=\binom{n}{k}p^{k}(1 - p)^{n - k} \), where \( \binom{n}{k}=\frac{n!}{k!(n - k)!} \).

Step2: Calculate the combination

First, calculate \( \binom{6}{2}=\frac{6!}{2!(6 - 2)!}=\frac{6!}{2!4!}=\frac{6\times5}{2\times1}=15 \).

Step3: Calculate the probability

Then, \( p^{k}=(0.56)^{2}=0.3136 \) and \( (1 - p)^{n - k}=(1 - 0.56)^{6 - 2}=(0.44)^{4}=0.44\times0.44\times0.44\times0.44 = 0.03748096 \).

Multiply these together: \( P(X = 2)=15\times0.3136\times0.03748096 \).

First, \( 15\times0.3136 = 4.704 \). Then, \( 4.704\times0.03748096\approx0.176 \).

Answer:

\( 0.176 \)