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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 41 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. \( P(X=41)=0.0851 \)

Ask by Page Morrison. in the United States
Mar 11,2025

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Answer

The probability that exactly 41 out of 50 small business owners do not have a college degree is 0.0851.

Solution

Let \( X \) be the number of small business owners without a college degree in a sample of 50. Since 76% do not have a college degree, we have \[ X \sim \text{Binomial}(n=50,\ p=0.76). \] The probability that exactly 41 do not have a college degree is given by the binomial probability formula: \[ P(X=41) = \binom{50}{41}(0.76)^{41}(0.24)^{50-41}. \] Since \[ \binom{50}{41} = \binom{50}{9}, \] we can write \[ P(X=41) = \binom{50}{9}(0.76)^{41}(0.24)^9. \] Evaluating this expression (using a calculator) gives \[ P(X=41) \approx 0.0851. \] Thus, the probability that exactly 41 out of 50 small business owners do not have a college degree is \[ \boxed{0.0851}. \]

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Beyond the Answer

Did you know that small business ownership doesn't necessarily require a college degree? In fact, many successful entrepreneurs thrive by leveraging their practical skills, industry knowledge, and networking abilities. This resilient spirit is a great reminder of how diverse paths can lead to success, often fueled by passion and experience rather than formal education. When it comes to probability calculations like the one you've presented, remember to use the binomial formula properly. For your example of 50 small business owners, you're dealing with a binomial distribution where \( n = 50 \) and \( p = 0.76 \). Just watch out for common mistakes, such as miscalculating the standard deviation or not rounding your calculations correctly. With practice, you'll be crunching these numbers like a pro!

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