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 dash nine 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.
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Part 1
​(a) Upper P left parenthesis 3 right parenthesisequals
  
enter your response here
​(Round to three decimal places as​ needed.)
Part 2
​(b) Upper P left parenthesis less than four right parenthesisequals
  
enter your response here
​(Round to three decimal places as​ needed.)
Part 3
​(c) Upper P left parenthesis at least three right parenthesisequals
  
enter your response here
​(Round to three decimal places as​ needed.)
Part 4
Which of the events are​ unusual? Select all that apply.
A.
The event Upper P left parenthesis 3 right parenthesis is unusual.
B.
The event Upper P left parenthesis less than four right parenthesis is unusual.
C.
The event Upper P left parenthesis at least three right parenthesis is unusual.
D.
None of the events are unusual.

To solve these problems, we'll use the **binomial distribution** because we are dealing with a fixed number of independent trials (7 U.S. adults) with two possible outcomes (either they oppose or do not oppose fracking), and a known probability of success (59% or 0.59).

### Binomial Distribution Formula
The binomial probability formula is:

\[
P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}
\]

Where:
- \( n = 7 \) (number of trials)
- \( p = 0.59 \) (probability of success)
- \( k \) is the number of successes we're interested in.

### Part 1: Probability of Exactly Three Opposing (P(3))
\[
P(X = 3) = \binom{7}{3} \cdot (0.59)^3 \cdot (0.41)^4
\]

### Part 2: Probability of Less than Four Opposing (P(X < 4))
\[
P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
\]

### Part 3: Probability of At Least Three Opposing (P(X \(\geq\) 3))
\[
P(X \geq 3) = 1 - P(X < 3)
\]
where \( P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) \).

Let's calculate these probabilities.

### Part 1: Probability of Exactly Three Opposing
\[
P(X = 3) = \binom{7}{3} \cdot (0.59)^3 \cdot (0.41)^4
\]
\[
\binom{7}{3} = \frac{7!}{3!(7-3)!} = 35
\]
\[
P(X = 3) = 35 \cdot (0.59)^3 \cdot (0.41)^4 \approx 35 \cdot 0.205379 \cdot 0.028305 = 0.203
\]

### Part 2: Probability of Less than Four Opposing
\[
P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
\]
Each can be calculated using the binomial formula, and adding up:

\[
P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) \approx 0.0004 + 0.0067 + 0.0456 + 0.203 = 0.255
\]

### Part 3: Probability of At Least Three Opposing
\[
P(X \geq 3) = 1 - P(X < 3) = 1 - (P(X = 0) + P(X = 1) + P(X = 2))
\]
\[
P(X \geq 3) \approx 1 - (0.0004 + 0.0067 + 0.0456) = 1 - 0.0527 = 0.947
\]

### Part 4: Determining if the Events are Unusual
An event is typically considered unusual if its probability is less than 0.05.

- **P(X = 3)** = 0.203 (Not unusual)
- **P(X < 4)** = 0.255 (Not unusual)
- **P(X \(\geq\) 3)** = 0.947 (Not unusual)

So, **None of the events are unusual**.

### Final Answers:
- **(a) P(3) = 0.203**
- **(b) P(X < 4) = 0.255**
- **(c) P(X \(\geq\) 3) = 0.947**
- **(d) None of the events are unusual.**

Would you like a deeper explanation on any of these parts or have any other questions?

### Relative Questions:
1. How is the binomial distribution different from the geometric distribution?
2. What is the significance of calculating probabilities for "at least" and "less than" scenarios?
3. How would the probabilities change if the number of adults surveyed increased?
4. Can the Poisson distribution be used in this context, and if so, how?
5. What does it mean for an event to be considered unusual in probability theory?
6. How does changing the probability of success (p) affect the binomial distribution?
7. Why might the binomial distribution be a more appropriate model than the normal distribution in this case?
8. How would you use technology (like a calculator or software) to find these probabilities?

**Tip:** When solving probability problems, make sure to first identify which distribution fits the scenario before starting your calculations.

Explore how to calculate probabilities using the binomial distribution for the scenario of U.S. adults opposing fracking. Learn about finding exact, less than, and at least probabilities. Detailed solutions and explanations included.

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