39​% adults favor the use of unmanned drones by police agencies. Twelve U.S. adults are randomly selected. Find the probability that the number of U.S. adults who favor the use of unmanned drones by police agencies is​ (a) exactly​ three, (b) at least​ four, (c) less than eight. 
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Part 1
​(a) Upper P left parenthesis 3 right parenthesisequals

This problem involves a **binomial probability distribution**, where we are given:

- The probability of success \( p = 0.39 \) (39% of adults favor the use of unmanned drones by police agencies).
- The probability of failure \( q = 1 - p = 0.61 \).
- The number of trials \( n = 12 \) (since 12 adults are selected).
- The random variable \( X \) represents the number of U.S. adults who favor the use of unmanned drones.

The binomial probability formula is given by:

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

Where:
- \( n = 12 \) is the number of trials (adults),
- \( k \) is the number of adults who favor drones,
- \( p = 0.39 \) is the probability of success (an adult favors drones),
- \( q = 1 - p = 0.61 \) is the probability of failure (an adult doesn't favor drones).

### Part (a) \( P(X = 3) \)
We are asked to find the probability that exactly 3 adults favor the use of drones. Using the binomial formula:

\[
P(X = 3) = \binom{12}{3} (0.39)^3 (0.61)^9
\]

Let's compute this.

The probability that exactly 3 U.S. adults out of 12 favor the use of unmanned drones by police agencies is approximately \( P(X = 3) = 0.153 \).

Would you like me to proceed with parts (b) and (c)? Let me know if you need further clarification on this part!

Here are 5 related questions to expand your understanding:
1. What is the binomial coefficient and how is it calculated?
2. How does changing the probability of success \(p\) affect the binomial probability distribution?
3. What would the probability be for more extreme values of \(X\), such as \(P(X = 0)\)?
4. How is the binomial distribution related to the normal distribution when \(n\) becomes large?
5. Can we apply this binomial formula for continuous data, or does it only apply to discrete events?

**Tip:** For large sample sizes, the binomial distribution can be approximated by the normal distribution if \( np \) and \( nq \) are both greater than 5.

39% of adults favor the use of unmanned drones by police agencies. Twelve U.S. adults are randomly selected. Find the probability that the number of U.S. adults who favor the use of unmanned drones by police agencies is (a) exactly three, (b) at least four, (c) less than eight.

Explore how to calculate binomial probabilities for a survey where 39% of adults favor the use of unmanned drones by police agencies. Learn to find the probability for exactly three, at least four, and less than eight adults out of twelve favoring drones. This detailed guide covers key probability concepts, suitable for students in Grades 10-12.

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