In Exercises 21–26, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities
Thirty-six percent of Americans think there is still a need for the practice of changing their clocks for Daylight Savings Time. You randomly select seven Americans. Find the probability that the number who say there is still a need for changing their clocks for Daylight Savings Time is (c) at least six.

Verified step by step guidance

1. Step 1: Identify the type of distribution to use. Since the problem involves a fixed number of trials (7 Americans), each with two possible outcomes (agree or disagree), and a constant probability of success (36% or 0.36), this is a binomial distribution problem.
2. Step 2: Define the parameters of the binomial distribution. The number of trials (n) is 7, the probability of success (p) is 0.36, and the number of successes (x) is at least 6. This means we are interested in P(X ≥ 6).
3. Step 3: Use the complement rule to simplify the calculation. P(X ≥ 6) can be rewritten as 1 - P(X ≤ 5). This allows us to calculate the cumulative probability for X ≤ 5 and subtract it from 1.
4. Step 4: Use the binomial probability formula to calculate P(X ≤ 5). The formula for the binomial probability is: P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where (n choose k) = n! / [k! * (n-k)!]. Compute this for k = 0, 1, 2, 3, 4, and 5, then sum these probabilities to find P(X ≤ 5).
5. Step 5: Subtract the cumulative probability P(X ≤ 5) from 1 to find P(X ≥ 6). Finally, determine whether the event is unusual by comparing the probability to a threshold (e.g., 0.05). If P(X ≥ 6) is less than 0.05, the event is considered unusual.

Key Concepts

Geometric Distribution

The geometric distribution models the number of trials needed to achieve the first success in a series of independent Bernoulli trials. It is characterized by a constant probability of success on each trial. In this context, it is not directly applicable since we are looking for the number of successes in a fixed number of trials, rather than the number of trials until the first success.

Binomial Distribution

The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this case, we can use the binomial distribution to find the probability of at least six Americans out of seven believing in the need for Daylight Savings Time, with the success probability being 0.36.

Unusual Events

An event is considered unusual if its probability is less than 0.05 (5%). In the context of this problem, after calculating the probability of at least six Americans supporting the practice of changing clocks, we can determine if this outcome is unusual by comparing the calculated probability to this threshold.