Test - questions

You have a test with eight questions, where you can choose from 3 answers for each question, and one answer is always correct. The probability that we answer 5 or 6 questions correctly when randomly filling in (that is, we all guess the answers) is ……. The average number of correctly guessed questions is ……. . (The quantity has a Binomial distribution of Bi, and the parameters are: the number of questions and probability that I will answer correctly).

Final Answer:

p =  0.0384

Step-by-step explanation:

$$\begin{gathered} C_5(7)=\left(\genfrac{}{}{0ex}{}{7}{5}\right)=\frac{7!}{5!(7-5)!}=\frac{7\cdot 6}{2\cdot 1}=21 \\ {p}_1=1/3=\frac{1}{3}\doteq 0.3333 \\ {p}_2=1-p_1=1-0.3333=\frac{2}{3}\doteq 0.6667 \\ p=\left(\genfrac{}{}{0ex}{}{5}{8-1}\right)\cdot p_1^5\cdot p_2^{8-6}=\left(\genfrac{}{}{0ex}{}{5}{8-1}\right)\cdot 0.3333^5\cdot 0.6667^{8-6}=0.0384 \end{gathered}$$

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