This problem involves a binomial distribution because we are dealing with a fixed number of trials (11 adults), and each trial (adult) results in either "success" (having very little confidence) or "failure" (not having very little confidence).

The formula for the binomial probability is:
$P(X = k) = \binom{n}{k} \cdot p^k \cdot (1 - p)^{n - k}$ Where:

• $n$ = number of trials = 11
• $k$ = number of successes (the specific number we are interested in)
• $p$ = probability of success (39% = 0.39)
• $\binom{n}{k} = \frac{n!}{k! \cdot (n - k)!}$ is the binomial coefficient.

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(a) Exactly 7 adults have very little confidence

$P(X = 7) = \binom{11}{7} \cdot (0.39)^7 \cdot (1 - 0.39)^4$ First, calculate the binomial coefficient: $\binom{11}{7} = \frac{11!}{7! \cdot (11 - 7)!} = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} = 330$

Now calculate the powers of probabilities:

• $0.39^7 \approx 0.002764$
• $0.61^4 \approx 0.138458$

Now multiply everything together: $P(X = 7) = 330 \cdot 0.002764 \cdot 0.138458 \approx 0.126$

So, the probability that exactly 7 adults have very little confidence is: $\boxed{0.126}$

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(b) Exactly 3 adults have very little confidence

$P(X = 3) = \binom{11}{3} \cdot (0.39)^3 \cdot (1 - 0.39)^8$ First, calculate the binomial coefficient: $\binom{11}{3} = \frac{11!}{3! \cdot (11 - 3)!} = \frac{11 \times 10 \times 9}{3 \times 2 \times 1} = 165$

Now calculate the powers of probabilities:

• $0.39^3 \approx 0.059319$
• $0.61^8 \approx 0.052922$

Now multiply everything together: $P(X = 3) = 165 \cdot 0.059319 \cdot 0.052922 \approx 0.517$

So, the probability that exactly 3 adults have very little confidence is: $\boxed{0.517}$

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Let me know if you need more steps or details!

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Follow-up Questions:

1. What is the mean and variance of the binomial distribution in this problem?
2. How would the probabilities change if the sample size increased to 20 adults?
3. What is the probability that fewer than 5 adults have very little confidence?
4. Can you use the normal approximation to the binomial distribution here? Why or why not?
5. What is the cumulative probability of having between 2 and 6 adults with little confidence?

Tip: When working with binomial probabilities, using a calculator or software helps avoid rounding errors.