To determine the probability that exactly 6 out of 10 city workers take the bus to work, we can make use of the binomial probability formula. The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials (in this case, each trial is whether or not a city worker takes the bus).
Given values:
- Total number of trials (n): 10
- Number of successes (k): 6
- Probability of success on a single trial (p): 0.15
The binomial probability formula is given by:
[tex]\[
P(k \text{ successes }) = { }_n C_k \cdot p^k \cdot (1-p)^{n-k}
\][/tex]
Here, [tex]\({ }_n C_k\)[/tex] is the binomial coefficient and is calculated as:
[tex]\[
{ }_n C_k = \frac{n!}{k!(n-k)!}
\][/tex]
Inserting the given values into the binomial coefficient formula:
[tex]\[
{ }_{10} C_6 = \frac{10!}{6!(10-6)!} = \frac{10!}{6! \cdot 4!}
\][/tex]
Next, evaluate [tex]\({ }_{10} C_6\)[/tex]:
[tex]\[
\frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210
\][/tex]
Now, use the binomial probability formula:
[tex]\[
P(6 \text{ successes }) = 210 \cdot (0.15)^6 \cdot (0.85)^{4}
\][/tex]
We can break this down into manageable steps:
1. Calculate [tex]\((0.15)^6\)[/tex]:
[tex]\[
(0.15)^6 \approx 0.00001139
\][/tex]
2. Calculate [tex]\((0.85)^4\)[/tex]:
[tex]\[
(0.85)^4 \approx 0.52200625
\][/tex]
3. Multiply these results by the binomial coefficient:
[tex]\[
P(6 \text{ successes }) = 210 \cdot 0.00001139 \cdot 0.52200625 \approx 0.0012486552626953104
\][/tex]
Finally, rounding this probability to the nearest thousandth:
[tex]\[
0.001
\][/tex]
Therefore, the probability that exactly 6 out of 10 city workers take the bus to work is approximately [tex]\(0.001\)[/tex]. The correct answer is [tex]\(0.001\)[/tex].
