Question

question number 4. (5.00 points) among 9 electrical components exactly one is known not to function properly. if 2 components are selected randomly, find the probability that exactly one does not function properly. 0.1111 0.8889 0.7023 0.2222 0.7778 none of the above

Explanation:

Step1: Identify the binomial - probability formula

The binomial - probability formula is $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success in a single trial, and $C(n,k)=\frac{n!}{k!(n - k)!}$. Here, $n = 2$ (the number of components selected), $k = 1$ (the number of non - functioning components), and the probability that a component is non - functioning is $p=\frac{1}{9}\approx0.1111$, and $1 - p=\frac{8}{9}\approx0.8889$.

Step2: Calculate the combination $C(n,k)$

$C(2,1)=\frac{2!}{1!(2 - 1)!}=\frac{2!}{1!1!}=\frac{2\times1!}{1!×1}=2$.

Step3: Calculate the probability $P(X = 1)$

$P(X = 1)=C(2,1)\times p^{1}\times(1 - p)^{2 - 1}=2\times\frac{1}{9}\times\frac{8}{9}=\frac{16}{81}\approx0.1975$. Since this is not among the given options, we made a wrong start. Let's consider another approach.
The probability that the first component is non - functioning and the second is functioning is $\frac{1}{9}\times\frac{8}{9}=\frac{8}{81}$, and the probability that the first component is functioning and the second is non - functioning is $\frac{8}{9}\times\frac{1}{9}=\frac{8}{81}$.
The total probability that exactly one is non - functioning is $\frac{8}{81}+\frac{8}{81}=\frac{16}{81}\approx0.1975$. Since this is not in the options, we calculate as follows:
The probability of choosing one non - working and one working component:
The number of ways to choose 2 components out of 9 is $C(9,2)=\frac{9!}{2!(9 - 2)!}=\frac{9\times8}{2\times1}=36$.
The number of ways to choose 1 non - working (1 out of 1 non - working) and 1 working (8 out of 8 working) is $C(1,1)\times C(8,1)=1\times8 = 8$.
The probability $P=\frac{C(1,1)\times C(8,1)}{C(9,2)}=\frac{8}{36}=\frac{2}{9}\approx0.2222$.

Answer:

0.2222