To solve the problems, we use the binomial distribution formula, where the probability of exactly $k$ successes in $n$ trials is given by:

$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

For cumulative probabilities, we sum up the probabilities for all values up to the desired $k$. Alternatively, a normal approximation can be used when $n$ is large and $p$ is not too close to 0 or 1.

Given:

• $n = 100$ (sample size),
• $p$ values are provided for each question,
• We'll calculate each part step-by-step.

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(a) Probability fifty or more think protecting the environment should be given priority

• $p = 0.51$,
• We need $P(X \geq 50)$,
• Use the complement rule: $P(X \geq 50) = 1 - P(X \leq 49)$.

(b) Probability forty or fewer think developing energy supplies is more important

• $p = 0.34$,
• We need $P(X \leq 40)$,
• Directly compute cumulative binomial or use normal approximation.

(c) Probability two or fewer have no opinion

• $p = 0.11$ (remaining $11\%$ with no opinion),
• We need $P(X \leq 2)$.

Let's compute these binom

Parameters for part (a)

n_a = 100 p_a = 0.51 x_a = 49 # Complement of 50 or more

Complementary cumulative probability

prob_a = 1 - binom.cdf(x_a, n_a, p_a)

Parameters for part (b)

n_b = 100 p_b = 0.34 x_b = 40 # 40 or fewer

Cumulative probability

prob_b = binom.cdf(x_b, n_b, p_b)

Parameters for part (c)

n_c = 100 p_c = 0.11 x_c = 2 # Two or fewer

Cumulative probability

prob_c = binom.cdf(x_c, n_c, pHere are the computed probabilities:

(a) The probability that fifty or more think protecting the environment should be given priority: 0.6181.

(b) The probability that forty or fewer think developing energy supplies is more important: 0.9137.

(c) The probability that two or fewer have no opinion: 0.0008.

Let me know if you want detailed steps for these calculations or have any questions.

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Related Questions:

1. How would the results change if the sample size increased to 1,000?
2. What is the standard deviation for each case in this problem?
3. How does the complement rule work in probability?
4. How would a normal approximation be used for part (a)?
5. What is the probability that exactly 50 people think the environment should be prioritized?

Tip:

When working with probabilities for large samples, consider using normal approximation to