Probability 81685

There are 49 products in the box, of which only 6 are good. We will randomly draw 6 products from them. What is the probability that at least four of the products drawn are good?

Final Answer:

p =  0.0027

Step-by-step explanation:

$$\begin{gathered} p_0=\frac{6}{49}\doteq 0.1224 \\ {p}_4=\left(\genfrac{}{}{0ex}{}{6}{4}\right)\cdot p_0^4\cdot (1-p_0{)}^{6-4}=(\genfrac{}{}{0ex}{}{6}{4})\cdot 0.1224^4\cdot (1-0.1224{)}^{6-4}=15\cdot 0.1224^4\cdot 0.8776^2\doteq 0.0026 \\ {p}_5=\left(\genfrac{}{}{0ex}{}{6}{5}\right)\cdot p_0^5\cdot (1-p_0{)}^{6-5}=(\genfrac{}{}{0ex}{}{6}{5})\cdot 0.1224^5\cdot (1-0.1224{)}^{6-5}=6\cdot 0.1224^5\cdot 0.8776^1\doteq 0.0001 \\ {p}_6=\left(\genfrac{}{}{0ex}{}{6}{6}\right)\cdot p_0^6\cdot (1-p_0{)}^{6-6}=(\genfrac{}{}{0ex}{}{6}{6})\cdot 0.1224^6\cdot (1-0.1224{)}^{6-6}=1\cdot 0.1224^6\cdot 0.8776^0\doteq 3.3708\cdot 10^{-6} \\ p=p_4+p_5+p_6=0.0026+0.0001+3.3708\cdot 10^{-6}=0.0027 \end{gathered}$$

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