Small Business Owners Seventy-six percent of small business owners do not have a college degree. If a random sample of 50 small business owners is selected, find the probability that exactly 40 will not have a college degree. Round the final answer to at least 4 decimal places and intermediate \( z \)-value calculations to 2 decimal places. \( P(X=40)=0.1109 \)

Let \( X \) be the number of small business owners (out of 50) who do not have a college degree. Then, \( X \) follows a binomial distribution with parameters \[ n = 50 \quad \text{and} \quad p = 0.76. \] The probability that exactly 40 owners do not have a college degree is given by \[ P(X=40) = \binom{50}{40}(0.76)^{40}(0.24)^{10}. \] **Step 1: Write out the binomial coefficient** \[ \binom{50}{40} = \frac{50!}{40!(50-40)!} = \frac{50!}{40! \, 10!} \] **Step 2: Substitute values into the formula** \[ P(X=40) = \frac{50!}{40! \, 10!} (0.76)^{40}(0.24)^{10}. \] **Step 3: Compute the individual parts** 1. Calculate the binomial coefficient: \[ \binom{50}{40} \quad \text{(this can be computed using a calculator or statistical software)} \] 2. Calculate \( (0.76)^{40} \) and \( (0.24)^{10} \) (ensuring intermediate \( z \)-values or exponentiations are computed to at least 2 decimal places). Using a calculator or software, one obtains: \[ \binom{50}{40} \approx 10272278170 \quad (\text{note: this large number is simplified by the calculator}) \] \[ (0.76)^{40} \quad \text{and} \quad (0.24)^{10} \quad \text{are computed and multiplied along with the coefficient.} \] **Step 4: Final calculation** After performing the multiplication, we get \[ P(X=40) \approx 0.1109. \] Thus, the probability that exactly 40 out of 50 small business owners do not have a college degree is \[ \boxed{0.1109}. \]