Question

A poll is given, showing 65% are in favor of a new building project.

If 8 people are chosen at random, what is the probability that exactly 7 of them favor the new building project?

Answer 1

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:
- \( n = 8 \)
- \( k = 7 \)
- \( p = 0.65 \)

1. **Calculate the binomial coefficient \( \binom{8}{7} \)**:

\[
\binom{8}{7} = \frac{8!}{7!(8-7)!} = \frac{8!}{7! \cdot 1!} = \frac{8 \cdot 7!}{7! \cdot 1} = 8
\]

2. **Calculate \( p^k \)**:

\[
p^7 = (0.65)^7
\]

Calculating \( (0.65)^7 \):

\[
(0.65)^7 \approx 0.04796
\]

3. **Calculate \( (1-p)^{n-k} \)**:

\[
(1-p)^{8-7} = (0.35)^1 = 0.35
\]

4. **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:

\[
\boxed{0.1343}
\]