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 (b) less than two

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 less than 2. This means we are interested in P(X < 2), which is the sum of probabilities for P(X = 0) and P(X = 1).
3. Step 3: Use the binomial probability formula to calculate P(X = 0) and P(X = 1). The formula is: P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where (n choose k) = n! / [k! * (n-k)!]. Substitute the values for k = 0 and k = 1 to compute these probabilities.
4. Step 4: Add the probabilities from Step 3 to find P(X < 2). Specifically, P(X < 2) = P(X = 0) + P(X = 1).
5. Step 5: Determine whether the event is unusual. An event is typically considered unusual if its probability is less than 0.05. Compare the calculated probability P(X < 2) to 0.05 to make this determination.

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 selecting fewer than two Americans who believe in changing clocks for Daylight Savings Time, given a probability of 0.36 and a sample size of 7.

Unusual Events

An event is considered unusual if its probability is less than 0.05 (5%). In the context of the binomial distribution, after calculating the probability of fewer than two successes, we can determine if this outcome is unusual by comparing the result to this threshold. This helps in understanding the significance of the observed data in relation to the expected probabilities.