Question

A poll is given, showing 25% are in favor of a new building project.

If 10 people are chosen at random, what is the probability that exactly 1 of them favor the new building project?

Answer 1

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:

1. **Calculate the binomial coefficient** \(\binom{10}{1}\):
\[
\binom{10}{1} = \frac{10!}{1!(10-1)!} = \frac{10!}{1! \cdot 9!} = \frac{10 \cdot 9!}{1 \cdot 9!} = 10
\]

2. **Calculate \( p^k \)** where \( k = 1 \):
\[
p^k = 0.25^1 = 0.25
\]

3. **Calculate \( (1 - p)^{n-k} \)** where \( n - k = 10 - 1 = 9 \):
\[
(1 - p)^{n-k} = 0.75^9
\]

Using a calculator, we find:
\[
0.75^9 \approx 0.0753
\]

4. **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:

\[
\boxed{0.1892}
\]