Seeds

From a box of spruce seeds with the germination of 80%, we randomly select 10 seeds and plant them. Find the median of the random variable: the number of germinating seeds.

Correct answer:

m =  8

Step-by-step explanation:

$$\begin{gathered} q=80\mathrm{\%}={\displaystyle \frac{80}{100}}=0.8 \\ r=1-q=1-0.8={\displaystyle \frac{1}{5}}=0.2 \\ n=10 \\ {n}_1=\left(\genfrac{}{}{0px}{}{n}{1}\right)\cdot q^1\cdot (1-q{)}^{n-1}=\left(\genfrac{}{}{0px}{}{10}{1}\right)\cdot 0.8^1\cdot (1-0.8{)}^{10-1}\doteq 4.096\cdot 10^{-6} \\ {n}_2=\left(\genfrac{}{}{0px}{}{n}{2}\right)\cdot q^2\cdot (1-q{)}^{n-2}=\left(\genfrac{}{}{0px}{}{10}{2}\right)\cdot 0.8^2\cdot (1-0.8{)}^{10-2}\doteq 0.0001 \\ {n}_3=\left(\genfrac{}{}{0px}{}{n}{3}\right)\cdot q^3\cdot (1-q{)}^{n-3}=\left(\genfrac{}{}{0px}{}{10}{3}\right)\cdot 0.8^3\cdot (1-0.8{)}^{10-3}\doteq 0.0008 \\ {n}_4=\left(\genfrac{}{}{0px}{}{n}{4}\right)\cdot q^4\cdot (1-q{)}^{n-4}=\left(\genfrac{}{}{0px}{}{10}{4}\right)\cdot 0.8^4\cdot (1-0.8{)}^{10-4}\doteq 0.0055 \\ {n}_5=\left(\genfrac{}{}{0px}{}{n}{5}\right)\cdot q^5\cdot (1-q{)}^{n-5}=\left(\genfrac{}{}{0px}{}{10}{5}\right)\cdot 0.8^5\cdot (1-0.8{)}^{10-5}\doteq 0.0264 \\ {n}_6=\left(\genfrac{}{}{0px}{}{n}{6}\right)\cdot q^6\cdot (1-q{)}^{n-6}=\left(\genfrac{}{}{0px}{}{10}{6}\right)\cdot 0.8^6\cdot (1-0.8{)}^{10-6}\doteq 0.0881 \\ {n}_7=\left(\genfrac{}{}{0px}{}{n}{7}\right)\cdot q^7\cdot (1-q{)}^{n-7}=\left(\genfrac{}{}{0px}{}{10}{7}\right)\cdot 0.8^7\cdot (1-0.8{)}^{10-7}\doteq 0.2013 \\ {n}_8=\left(\genfrac{}{}{0px}{}{n}{8}\right)\cdot q^8\cdot (1-q{)}^{n-8}=\left(\genfrac{}{}{0px}{}{10}{8}\right)\cdot 0.8^8\cdot (1-0.8{)}^{10-8}\doteq 0.302 \\ {n}_9=\left(\genfrac{}{}{0px}{}{n}{9}\right)\cdot q^9\cdot (1-q{)}^{n-9}=\left(\genfrac{}{}{0px}{}{10}{9}\right)\cdot 0.8^9\cdot (1-0.8{)}^{10-9}\doteq 0.2684 \\ {n}_{10}=\left(\genfrac{}{}{0px}{}{n}{10}\right)\cdot q^{10}\cdot (1-q{)}^{n-10}=\left(\genfrac{}{}{0px}{}{10}{10}\right)\cdot 0.8^{10}\cdot (1-0.8{)}^{10-10}\doteq 0.1074 \\ {n}_8>\mathrm{\forall }{n}_i \\ m=8 \end{gathered}$$

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