Let's break down the problem and solve it using the binomial distribution, since the scenario involves a fixed number of trials (7 adults), two possible outcomes (oppose or not oppose fracking), and a fixed probability of success (opposing fracking).

Given dataLet's break down the problem and solve it using the binomial distribution, since the scenario involves a fixed number of trials (7 adults), two possible outcomes (oppose or not oppose fracking), and a fixed probability of success (opposing fracking).

Given

• Probability of success $p = 0.59$
• Number of trials $n = 7$
• We will calculate the probabilities for (a) exactly 3, (b) less than 4, and (c) at least 3 adults opposing fracking.

Part 1: $P(X = 3)$

To calculate the probability of exactly 3 adults opposing fracking, we use the binomial probability formula: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$ where:

• $n = 7$
• $k = 3$
• $p = 0.59$
• $1-p = 0.41$

$P(X = 3) = \binom{7}{3} (0.59)^3 (0.41)^4$

Calculating this:

$P(X = 3) \approx 0.203$

Part 2: $P(X < 4)$

To find the probability that fewer than 4 adults oppose fracking, we calculate $P(X < 4)$, which is the sum of the probabilities for $X = 0$, $X = 1$, $X = 2$, and $X = 3$:

$P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)$

This can be calculated as follows:

• $P(X = 0) \approx 0.002$
• $P(X = 1) \approx 0.020$
• $P(X = 2) \approx 0.097$
• $P(X = 3) \approx 0.203$

$P(X < 4) \approx 0.002 + 0.020 + 0.097 + 0.203 = 0.322$

Part 3: $P(X \geq 3)$

To find the probability that at least 3 adults oppose fracking, we calculate $P(X \geq 3)$, which is the complement of $P(X < 3)$:

$P(X \geq 3) = 1 - P(X < 3)$

We already have $P(X < 3)$:

$P(X < 3) \approx 0.002 + 0.020 + 0.097 = 0.119$

$P(X \geq 3) = 1 - 0.119 = 0.881$

Part 4: Which of the events are unusual?

An event is typically considered unusual if its probability is less than 0.05.

• $P(X = 3) \approx 0.203$ (Not unusual)
• $P(X < 4) \approx 0.322$ (Not unusual)
• $P(X \geq 3) \approx 0.881$ (Not unusual)

Thus, the correct answer is: D. None of the events are unusual.

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Do you have any questions or would you like more details?

8 Related Questions:

1. How do you calculate the mean and variance of a binomial distribution?
2. What is the difference between the binomial and Poisson distributions?
3. How do you interpret a probability of 0.05 in hypothesis testing?
4. How does the probability of success affect the shape of a binomial distribution?
5. What conditions make an event unusual in probability theory?
6. Can the binomial distribution be used for large sample sizes?
7. How do you calculate cumulative probabilities for a binomial distribution?
8. What is the significance of the "complement rule" in probability?

Tip: Always check whether the binomial distribution assumptions apply before using it to model a real-world scenario.