Test

The teacher prepared a test with ten questions. The student can choose one correct answer from the four (A, B, C, D). The student did not get a written exam at all. What is the probability that:

a) He answers half correctly.
b) He answers all correctly.
c) He does not answer any question correctly.

Final Answer:

a =  0.0584
b =  0
c =  0.0563

Step-by-step explanation:

$$\begin{gathered} C_5(10)=\left(\genfrac{}{}{0ex}{}{10}{5}\right)=\frac{10!}{5!(10-5)!}=\frac{10\cdot 9\cdot 8\cdot 7\cdot 6}{5\cdot 4\cdot 3\cdot 2\cdot 1}=252 \\ p=1/4=\frac{1}{4}=0.25 \\ q=1-p=1-0.25=\frac{3}{4}=0.75 \\ n=10 \\ {n}_1=n/2=10/2=5 \\ a=\left(\genfrac{}{}{0ex}{}{n}{n_1}\right)\cdot p^{n_1}\cdot q^{n-n_1}=252\cdot 0.25^5\cdot 0.75^{10-5}=0.0584 \end{gathered}$$

$$\begin{gathered} C_{10}(10)=\left(\genfrac{}{}{0ex}{}{10}{10}\right)=\frac{10!}{10!(10-10)!}=\frac{1}{1}=1 \\ b=\left(\genfrac{}{}{0ex}{}{n}{n}\right)\cdot p^n\cdot q^{n-n}=1\cdot 0.25^{10}\cdot 0.75^{10-10}=0 \end{gathered}$$

$$\begin{gathered} C_0(10)=\left(\genfrac{}{}{0ex}{}{10}{0}\right)=\frac{10!}{0!(10-0)!}=\frac{1}{1}=1 \\ c=\left(\genfrac{}{}{0ex}{}{n}{0}\right)\cdot p^0\cdot q^{n-0}=1\cdot 0.25^0\cdot 0.75^{10-0}=0.0563 \end{gathered}$$

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