Answer: 0.3756
Step-by-step explanation:
Given : A biotechnology company produced 196 doses of somatropin, including 9 which were defective.
Proportion of defective doses of somatropin : [tex]p=\dfrac{9}{196}=0.0459183673469\approx0.046[/tex]
Using binomial probability formula,
[tex]P(x)=^nC_xp^x(1-p)^{n-x}[/tex]
If Quality control test 10 samples at random, and rejects the batch if any of the random samples are found defective.
Then, the probability that the batch gets rejected will be :-
[tex]P(x\geq1)=1-P(x=0)\\\\=1-^{10}C_0(0.046)^0(1-0.046)^{10}\\\\=1-(1-0.046)^{10}\\\\=1-0.624429981287\\\\=0.375570018713\approx0.3756[/tex]
Hence, the probability that the batch gets rejected = 0.3756
Using the hypergeometric distribution, it is found that there is a 0.3821 = 38.21% probability that the batch gets rejected.
The samples are chosen without replacement, hence the hypergeometric distribution is used to solve this question.
The formula is:
[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
The parameters are:
In this problem:
The probability that the batch gets reject, that is, the probability that there is at least one defective, is given by:
[tex]P(X \geq 1) = 1 - P(X = 0)[/tex]
In which:
[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}[/tex]
[tex]P(X = 0) = h(0,196,10,9) = \frac{C_{9,0}C_{187,10}}{C_{196,10}} = 0.6179[/tex]
Then:
[tex]P(X \geq 1) = 1 - P(X = 0) = 1 - 0.6179 = 0.3821[/tex]
0.3821 = 38.21% probability that the batch gets rejected.
More can be learned about the hypergeometric distribution at https://brainly.com/question/24826394
