Probability of failures

In certain productions, the probability of failures is 0.01. Calculate the probability that there will be more than 1 failure among the 100 selected products if we return the selected products to the file after the check.

Correct result:

p =  0.2642

Solution:

$$\begin{gathered} C_0(100)=\left({\displaystyle \genfrac{}{}{0px}{}{100}{0}}\right)={\displaystyle \frac{100!}{0!(100-0)!}}={\displaystyle \frac{1}{1}}=1 \\ {C}_1(100)=\left({\displaystyle \genfrac{}{}{0px}{}{100}{1}}\right)={\displaystyle \frac{100!}{1!(100-1)!}}={\displaystyle \frac{100}{1}}=100 \\ q=0.01 \\ n=100 \\ {p}_0=\left(\genfrac{}{}{0px}{}{n}{0}\right)\cdot q^0\cdot (1-q{)}^{n-0}=1\cdot 0.01^0\cdot (1-0.01{)}^{100-0}\doteq 0.366 \\ {p}_1=\left(\genfrac{}{}{0px}{}{n}{1}\right)\cdot q^1\cdot (1-q{)}^{n-1}=100\cdot 0.01^1\cdot (1-0.01{)}^{100-1}\doteq 0.3697 \\ p=1-(p_0+p_1)=1-(0.366+0.3697)=0.2642 \end{gathered}$$

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