To determine the probability that exactly 1 out of 10 randomly chosen people favors a new building project, we can use the binomial probability formula. The formula is given by:
\[ P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k} \]
Where:
- \( n \) is the total number of trials (in this case, \( n = 10 \)),
- \( k \) is the number of successful outcomes we want (in this case, \( k = 1 \)),
- \( p \) is the probability of success on a single trial (in this case, \( p = 0.25 \)),
- \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \).
Let's calculate it step by step:
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Calculate the binomial coefficient \(\binom{10}{1}\): \[ \binom{10}{1} = \frac{10!}{1!(10-1)!} = \frac{10!}{1! \cdot 9!} = \frac{10 \cdot 9!}{1 \cdot 9!} = 10 \]
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Calculate \( p^k \) where \( k = 1 \): \[ p^k = 0.25^1 = 0.25 \]
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Calculate \( (1 - p)^{n-k} \) where \( n - k = 10 - 1 = 9 \): \[ (1 - p)^{n-k} = 0.75^9 \]
Using a calculator, we find: \[ 0.75^9 \approx 0.0753 \]
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Put it all together to find \( P(X = 1) \): \[ P(X = 1) = \binom{10}{1} (0.25)^1 (0.75)^9 \] \[ P(X = 1) = 10 \times 0.25 \times 0.0753 \] \[ P(X = 1) = 2.5 \times 0.0753 \approx 0.18925 \]
Thus, the probability that exactly 1 out of 10 people favors the new building project is approximately:
\[ \boxed{0.1892} \]
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