Exam pass probability

During the exam, each student receives 30 different questions, from which he chooses 3 at random. To pass the exam, he needs to be able to answer two correctly. What is the probability that a student will pass if he mastered 70% of the questions (70% of the questions are taught)?

Final Answer:

p =  0.784

Step-by-step explanation:

$$\begin{gathered} n=30 \\ {n}_1=3 \\ {n}_2=2 \\ {p}_0=70\%=\frac{70}{100}=\frac{7}{10}=0.7 \\ {p}_2=\left(\genfrac{}{}{0ex}{}{3}{2}\right)\cdot p_0^2\cdot (1-p_0{)}^{3-2}=(\genfrac{}{}{0ex}{}{3}{2})\cdot 0.7^2\cdot (1-0.7{)}^{3-2}=3\cdot 0.7^2\cdot 0.3^1=\frac{441}{1000}=0.441 \\ {p}_3=\left(\genfrac{}{}{0ex}{}{3}{3}\right)\cdot p_0^3\cdot (1-p_0{)}^{3-3}=(\genfrac{}{}{0ex}{}{3}{3})\cdot 0.7^3\cdot (1-0.7{)}^{3-3}=1\cdot 0.7^3\cdot 0.3^0=\frac{343}{1000}=0.343 \\ p=p_2+p_3=0.441+0.343=0.784 \end{gathered}$$

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