Exam question probability

During the exam, the student takes 3 questions out of 20. He is ready for 14 of them. Find the probability that he draws at least one that he knows.

Final Answer:

p =  0.973

Step-by-step explanation:

$$\begin{gathered} r=\frac{14}{20}=\frac{7}{10}=0.7 \\ {p}_1=\left(\genfrac{}{}{0ex}{}{3}{1}\right)\cdot r^1\cdot (1-r{)}^{3-1}=(\genfrac{}{}{0ex}{}{3}{1})\cdot 0.7^1\cdot (1-0.7{)}^{3-1}=3\cdot 0.7^1\cdot 0.3^2=\frac{189}{1000}=0.189 \\ {p}_2=\left(\genfrac{}{}{0ex}{}{3}{2}\right)\cdot r^2\cdot (1-r{)}^{3-2}=(\genfrac{}{}{0ex}{}{3}{2})\cdot 0.7^2\cdot (1-0.7{)}^{3-2}=3\cdot 0.7^2\cdot 0.3^1=\frac{441}{1000}=0.441 \\ {p}_3=\left(\genfrac{}{}{0ex}{}{3}{3}\right)\cdot r^3\cdot (1-r{)}^{3-3}=(\genfrac{}{}{0ex}{}{3}{3})\cdot 0.7^3\cdot (1-0.7{)}^{3-3}=1\cdot 0.7^3\cdot 0.3^0=\frac{343}{1000}=0.343 \\ p=p_1+p_2+p_3=0.189+0.441+0.343=0.973 \end{gathered}$$

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