Probability - test

The test contains 11 questions, each question has 4 options (only one correct). Find the probability that less than 2 questions will be answered correctly. (The probability of a correctly answered question is 1/4.) (Write the result as a whole number in percent, round mathematically. For example, write the probability of 0.535 as 54%.)

Correct answer:

a =  20 %

Step-by-step explanation:

$$\begin{gathered} n=11 \\ p=1/4=\frac{1}{4}=0.25 \\ {p}_0=\left(\genfrac{}{}{0ex}{}{n}{0}\right)\cdot p^0\cdot (1-p{)}^{n-0}=(\genfrac{}{}{0ex}{}{11}{0})\cdot 0.25^0\cdot (1-0.25{)}^{11-0}=1\cdot 0.25^0\cdot 0.75^{11}\doteq 0.0422 \\ {p}_1=\left(\genfrac{}{}{0ex}{}{n}{1}\right)\cdot p^1\cdot (1-p{)}^{n-1}=(\genfrac{}{}{0ex}{}{11}{1})\cdot 0.25^1\cdot (1-0.25{)}^{11-1}=11\cdot 0.25^1\cdot 0.75^{10}\doteq 0.1549 \\ a=100\cdot (p_0+p_1)=100\cdot (0.0422+0.1549)=20\% \end{gathered}$$

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