The second sample will be necessary:
The packing machine is giving 80%=0.8 of items properly parked, unsatisfactory packages 1-0.8=0.2.
3 or more unsatisfactory packages: 3 or 4 or 5
Choose 3 from 5 unsatisfactory packages:
"p_1=\\frac{3}{5}\\cdot 0.2"
Choose 4 from 5 unsatisfactory packages:
"p_2=\\frac{4}{5}\\cdot0.2"
Choose 5 from 5 unsatisfactory packages:
"p_3=\\frac{5}{5}\\cdot0.2"
"P=p_1+p_2+p_3=\\frac{3}{5}\\cdot0.2+\\frac{4}{5}\\cdot0.2+\\frac{5}{5}\\cdot0.2=0.48"
The probability 0.48 that second sample will be necessary.
Out of the two samples (10 items), there are 9 satisfactory:
We use the Bernoulli formula:
"P=(\\begin{matrix}\n k \\\\\n n\n\\end{matrix})p^{k}q^{n-k}"
In our case
"p=0.8\\\\\nq=0.2\\\\\n( \\begin{matrix}\n 9\\\\\n 10\n\\end{matrix})=\\frac{10!}{9!(10-9)!}=\\frac{10\\cdot 9!}{9!\\cdot 1!}=10\\\\\nP=10\\cdot 0.8^9\\cdot0.2^{10-9}=10\\cdot 0.8^9\\cdot0.2^1\\approx 0.268"
The 0.268 probability that, out of the two samples (10 items), there are 9 satisfactory
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I am so happy that you have solved my math problem using a very clear method of solving binomial problems. This question is actually my assignment question which is due for handing in on wednesday and i was beginning to panic. Thank you very much and i will definitely recommend you to my colleagues. i am doing a masters degree in business administration with Finance. Thank you once again i appreciate it.
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