Error rate

The exam has six questions. Students have an error rate of 20% and can have a maximum of 1 question wrong. What is the probability that they will succeed?

Correct answer:

p =  0.6554

Step-by-step explanation:

$$\begin{gathered} n=6 \\ {p}_1=20\%=\frac{20}{100}=\frac{1}{5}=0.2 \\ r=1-p_1=1-0.2=\frac{4}{5}=0.8 \\ a=\left(\genfrac{}{}{0ex}{}{n}{6}\right)\cdot r^6\cdot (1-r{)}^{n-6}=(\genfrac{}{}{0ex}{}{6}{6})\cdot 0.8^6\cdot (1-0.8{)}^{6-6}=1\cdot 0.8^6\cdot 0.2^0\doteq 0.2621 \\ b=\left(\genfrac{}{}{0ex}{}{n}{5}\right)\cdot r^5\cdot (1-r{)}^{n-5}=(\genfrac{}{}{0ex}{}{6}{5})\cdot 0.8^5\cdot (1-0.8{)}^{6-5}=6\cdot 0.8^5\cdot 0.2^1\doteq 0.3932 \\ p=a+b=0.2621+0.3932=0.6554 \end{gathered}$$

Try another example