Probability

In a batch of 500 products, there are 10 rejects. In a statistical inspection, 40 products are checked, which are randomly selected from a batch, and the selection is made without repetition. The batch is considered satisfactory if there is at most one defective product among the 40 inspected products. Calculate the probability that the specified batch will be considered satisfactory.

Correct answer:

p =  0.955

Step-by-step explanation:

$$\begin{gathered} p_1=\frac{10}{500}=\frac{1}{50}=0.02 \\ {p}_2=\frac{1}{40}=0.025 \\ {C}_0(40)=\left(\genfrac{}{}{0ex}{}{40}{0}\right)=\frac{40!}{0!(40-0)!}=\frac{1}{1}=1 \\ {p}_3=\left(\genfrac{}{}{0ex}{}{40}{0}\right)\cdot p_1^0\cdot (1-p_1{)}^{40}=1\cdot {\frac{1}{50}}^0\cdot (1-\frac{1}{50}{)}^{40}\doteq 0.4457 \\ {C}_1(40)=\left(\genfrac{}{}{0ex}{}{40}{1}\right)=\frac{40!}{1!(40-1)!}=\frac{40}{1}=40 \\ {p}_4=\left(\genfrac{}{}{0ex}{}{40}{1}\right)\cdot p_1^1\cdot (1-p_1{)}^{39}=40\cdot {\frac{1}{50}}^1\cdot (1-\frac{1}{50}{)}^{39}\doteq 0.3638 \\ p=1-(p_1+p_2)=1-(\frac{1}{50}+\frac{1}{40})=\frac{191}{200}=0.955 \end{gathered}$$

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