 Brock Brown

2022-01-04

A batch of 140 semiconductor chips is inspected by choosing a sample of 5 chips. Assume 10 of the chips do not conform to customer requirements.
a. How many different samples are possible?
b. How many samples of five contain exactly one nonconforming chip?
c. How many samples of five contain at least one nonconforming chip? Lindsey Gamble

Step 1
Given:
$n=140$
$r=5$
10 of the 140 chips are non-conforming.
Definition permutation (order is important):
${P}_{n,r}=\frac{n!}{\left(n-r\right)!}$
Definition combination (order is not important):
${C}_{n,r}=\frac{n!}{r!\left(n-r\right)!}$
with $n\ne n\cdot \left(n-1\right)\cdot \cdots \cdot 2\cdot 1$
Step 2
a. Order is not important, because it is not important whether a chip was selected first or was selected last and thus we need to use the definition of a combination.
${C}_{140,5}=\frac{140!}{5!\left(140-5\right)!}=\frac{140!}{5!135!}=416,965,528$
Step 3
b. The sample contains 1 nonconforming chip, which means that 1 nonconforming chip was selected from the 10 nonconforming chips:
${C}_{10,1}=\frac{10!}{1!\left(10-1\right)!}=\frac{10!}{1!9!}=10$
The sample contains 1 nonconforming chip out of 5 chips and thus also contains 4 conforming chips, which means that 4 conforming chip was selected from the 130 conforming chips (140 chips in total of which 10 are nonconforming, thus 130 have to be conforming):
${C}_{130,4}=\frac{130!}{4!\left(130-4\right)!}=\frac{130!}{4!126!}=11,358,880$
The total number of samples in which to select a sample with 1 nonconfirming chip is then the product of the number of ways to select 1 nonconforming chip and the number of ways to select 4 conforming chips.
$10×11,358,880=113,588,800$
Step 4
c. Let us first determine the number of samples that contain no nonconforming chips. The number of samples with at least one nonconforming chips is then the total number of samples decreased by the number of samples with no conforming chips.
The sample contains 0 nonconforming chips, which means that 0 nonconforming chips was selected from the 10 nonconforming chips:
${C}_{10,0}=\frac{10!}{0!\left(10-0\right)!}=\frac{10!}{0!10!}=1$
The sample contains 0 nonconforming chips out of 5 chips and thus also contains 5 conforming chips, which means that 5 conforming chip was selected from the 130 conforming chips (140 chips in total of which 10 are nonconforming, thus 130 have to be conforming):
${C}_{130,4}=\frac{130!}{5!\left(130-5\right)!}=\frac{130!}{5!125!}=286,243,776$
The total number of samples in which to select a sample with 0 nonconforming chips is then the product of the number of ways to select 0 nonconforming chips and the number of ways to select 5 conforming chips.
$1×286,243,776=286,243,776$
The number of samples with at least one nonconforming chips is then the total number of samples (part a) decreased by the number of samples with no conforming chips.
$416,965,528-286,243,776=130,721,752$ mauricio0815sh

a) The lot size is 140, the sample size is 5
$\left(\begin{array}{c}140\\ 5\end{array}\right)=146,965,528$
b) How many samples contain 4 good and i bad chip:
$\left(\begin{array}{c}130\\ 4\end{array}\right)\left(\begin{array}{c}10\\ 1\end{array}\right)=113,588,800$
c) How many samples contain:
(4 good and 1 bad) or (3 good and 2 bad) or (2 good and 3 bad) or (1 good and 4 bad) or (0 good and 5 bad)
$=\left(\begin{array}{c}130\\ 4\end{array}\right)\left(\begin{array}{c}10\\ 1\end{array}\right)+\left(\begin{array}{c}130\\ 3\end{array}\right)\left(\begin{array}{c}10\\ 2\end{array}\right)+\left(\begin{array}{c}130\\ 2\end{array}\right)\left(\begin{array}{c}10\\ 3\end{array}\right)+\left(\begin{array}{c}150\\ 1\end{array}\right)\left(\begin{array}{c}10\\ 4\end{array}\right)+\left(\begin{array}{c}130\\ 0\end{array}\right)\left(\begin{array}{c}10\\ 5\end{array}\right)$
$=113,588,800+16,099,200+1,006,200+27,300+252$
$=130,721,752$ karton

Step 1
10 - do not confirm requirement.
130 - confirm requirement
140 - total semiconductor
select 5 at random.
a) No.of samples possible,
Step 2

b) How many of five contains exactly one nonconfirming

$=113588800$
c)  = Total wars - no nonconfirming
$={140}_{5}^{C}-{130}_{5}^{C}$
$=130721752$ karton