Who Do You Trust?
According to the National Constitution Center, 18% of Americans trust organized religion.


c. In a random sample of three Americans, all three indicated they trust organized religion. Is this result surprising?

Verified step by step guidance

1. Identify the probability that a single American trusts organized religion, which is given as \(p = 0.18\).
2. Since the sample consists of three Americans, and we want the probability that all three trust organized religion, recognize this as a binomial probability problem where the number of trials \(n = 3\) and the number of successes \(k = 3\).
3. Calculate the probability that all three trust organized religion using the formula for the probability of exactly \(k\) successes in \(n\) independent trials: \[P(X = k) = \binom{n}{k} p^{k} (1-p)^{n-k}\] In this case, since \(k = n = 3\), the formula simplifies to: \[P(X = 3) = p^{3}\]
4. Compute \(p^{3}\) by raising the probability \$0.18$ to the power of 3 to find the probability that all three trust organized religion.
5. Interpret the result: if this probability is very low (commonly less than 5%), then the event is considered surprising; otherwise, it is not surprising.

Key Concepts

Binomial Probability

Binomial probability calculates the likelihood of a specific number of successes in a fixed number of independent trials, each with the same probability of success. Here, 'success' means an individual trusting organized religion, with probability 0.18, and we want the probability all three in a sample trust it.

Independence of Events

Independence means the outcome of one event does not affect another. Assuming each American's trust is independent allows us to multiply individual probabilities to find the chance all three trust organized religion.

Interpreting Probability Results

After calculating the probability of all three trusting religion, we assess if the result is surprising by comparing it to common thresholds (e.g., very low probabilities indicate surprising outcomes). This helps determine if the observed sample is unusual.