According to a survey, 15% of city workers take the bus to work. Donatella randomly surveys 10 workers. What is the probability that exactly 6 workers take the bus to work? Round the answer to the nearest thousandth.
[tex]
\begin{aligned}
P(k \text { successes }) & ={{ }_n C_k p^k(1-p)^{n-k}} \\
{ }_n C_k & =\frac{n!}{(n-k)!\cdot k!}
\end{aligned}
[/tex]
Answer (1)
Use the binomial probability formula: P ( k ) = n C k p k ( 1 − p ) n − k .
Identify n = 10 , k = 6 , and p = 0.15 .
Calculate P ( 6 ) = 10 C 6 ( 0.15 ) 6 ( 0.85 ) 4 .
The probability is approximately 0.001 .
Explanation
Analyze the problem We are given that 15% of city workers take the bus to work. Donatella surveys 10 workers, and we want to find the probability that exactly 6 of them take the bus. This is a binomial probability problem.
State the binomial probability formula The binomial probability formula is given by: P ( k successes ) = n C k p k ( 1 − p ) n − k where:
n is the number of trials (workers surveyed)
k is the number of successes (workers who take the bus)
p is the probability of success (probability a worker takes the bus)
n C k is the binomial coefficient, calculated as ( n − k )! k ! n !
Identify the parameters In this case, we have:
n = 10 (10 workers surveyed)
k = 6 (exactly 6 workers take the bus)
p = 0.15 (15% of workers take the bus)
Calculate the binomial coefficient First, we calculate the binomial coefficient: 10 C 6 = ( 10 − 6 )! 6 ! 10 ! = 4 ! 6 ! 10 ! = 4 × 3 × 2 × 1 10 × 9 × 8 × 7 = 210 So, 10 C 6 = 210 .
Calculate p^k Next, we calculate p k :
( 0.15 ) 6 = 0.000011390625
Calculate (1-p)^(n-k) Then, we calculate ( 1 − p ) n − k :
( 1 − 0.15 ) 10 − 6 = ( 0.85 ) 4 = 0.52200625
Calculate the final probability Now, we plug these values into the binomial probability formula: P ( 6 successes ) = 10 C 6 × ( 0.15 ) 6 × ( 0.85 ) 4 = 210 × 0.000011390625 × 0.52200625 = 0.001249 Rounding to the nearest thousandth, we get 0.001.
State the final answer Therefore, the probability that exactly 6 out of 10 randomly surveyed city workers take the bus to work is approximately 0.001.
Examples
Consider a quality control scenario where a company produces items, and each item has a 15% chance of being defective. If you randomly inspect 10 items, the probability of finding exactly 6 defective items can be calculated using the binomial probability formula. This helps the company understand the likelihood of encountering such a situation and adjust their production process accordingly. The formula is also applicable in medical research, where you might want to determine the probability of a certain number of patients experiencing side effects from a drug, given the probability of a single patient experiencing the side effect.
