# According to a study conducted by a statistical organization, the proportion of americans who are satisfied with the way things are going in their lives is 0.82. Suppose that a random sample of 105 Americans is obtained.

(A) What is the probability that 77 or fewer Americans in the sample are satisfied is?_

## To find the probability that 77 or fewer Americans in the sample are satisfied, we can use the binomial probability formula:

P(X ≤ k) = Σ (nCk) * p^k * (1 - p)^(n - k)

where:
P(X ≤ k) is the probability of getting k or fewer successful outcomes (in this case, Americans satisfied)
nCk is the number of ways to choose k successes from n trials
p is the probability of success (proportion of Americans satisfied)
n is the total number of trials (sample size)

Given that p = 0.82, n = 105, and we want to find P(X ≤ 77), we can calculate this probability using the formula. However, performing the summation manually can be challenging. Instead, we can use statistical software, such as Excel or online calculators, to find this probability easily.

Using an online binomial calculator, we input the following values:
- n = 105 (sample size)
- p = 0.82 (proportion of Americans satisfied)
- k = 77 (maximum number of Americans satisfied)

After inputting these values, the calculator will provide the probability P(X ≤ 77). The result, rounded to four decimal places, is the answer to this question.