Solution:
Using the Bernoulli formula, we find the probabilities that 5 out of 5 and 4 out of 5 samples will be of high quality.
"p_{m,n}\\approx \\frac {1}{\\sqrt{npq}} \\varphi{(x)}"
"\\varphi {(x)}=\\frac {1}{\\sqrt{2 \\pi}} \\exp ^{\\frac {-x^2}{2}}"
"x=\\frac {m-np}{\\sqrt {npq}}"
Let m=5, n=5, p=0.8, q=0.2
"p_{5,5}\\approx 0.239"
Let m=4, n=5, p=0.8, q=0.2
"p_{4,5} \\approx 0.447"
Since either the first or second option is possible, the probability that a second measurement is not required is
The likelihood that a second check will be necessary will be 1-0.686=0.314
We find the probability that out of 10 packs, 9 will be packed correctly
"C_{10} ^9\\times 0.8^9\\times 0.2^1=0.053"
Since the machine produces fully packaged products in 80% of cases, the probability of receiving a packaged product is 0.8, therefore, the probability of an unpacked product is 1-0.8 = 0.2.
Answer: 0.314; 0.053
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