To solve this problem, we can use the binomial probability formula, which is given by:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]
where:
- \( n \) is the number of trials (in this case, 8 people),
- \( k \) is the number of successes (in this case, 7 people in favor),
- \( p \) is the probability of success on an individual trial (in this case, 0.65),
- \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \).
Substituting the values into the formula:
- \( n = 8 \)
- \( k = 7 \)
- \( p = 0.65 \)
- Calculate the binomial coefficient \( \binom{8}{7} \):
\[ \binom{8}{7} = \frac{8!}{7!(8-7)!} = \frac{8!}{7! \cdot 1!} = \frac{8 \cdot 7!}{7! \cdot 1} = 8 \]
- Calculate \( p^k \):
\[ p^7 = (0.65)^7 \]
Calculating \( (0.65)^7 \):
\[ (0.65)^7 \approx 0.04796 \]
- Calculate \( (1-p)^{n-k} \):
\[ (1-p)^{8-7} = (0.35)^1 = 0.35 \]
- Combine everything into the binomial formula:
Now substitute all of this into the binomial probability formula:
\[ P(X = 7) = \binom{8}{7} (0.65)^7 (0.35)^1 \]
So,
\[ P(X = 7) = 8 \cdot (0.65)^7 \cdot (0.35) \]
Calculating this,
\[ P(X = 7) = 8 \cdot 0.04796 \cdot 0.35 \approx 8 \cdot 0.016786 = 0.134288 \]
Thus, the probability that exactly 7 out of 8 randomly chosen people favor the new building project is approximately:
\[ \boxed{0.1343} \]
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