Test

The teacher prepared a test with ten questions. The student has the option to 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 either one answer correctly.

Correct result:

a =  0.058
b =  0
c =  0.056

Solution:

C5(10)=(105)=10!5!(105)!=10987654321=252 p=1/4=14=0.25 q=1p=10.25=34=0.75 n=10 n1=n/2=10/2=5 a=(nn1) pn1 qnn1=252 0.255 0.75105=0.058C_{{ 5}}(10) = \dbinom{ 10}{ 5} = \dfrac{ 10! }{ 5!(10-5)!} = \dfrac{ 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 } { 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 } = 252 \ \\ p=1/4=\dfrac{ 1 }{ 4 }=0.25 \ \\ q=1-p=1-0.25=\dfrac{ 3 }{ 4 }=0.75 \ \\ n=10 \ \\ n_{1}=n/2=10/2=5 \ \\ a={ { n } \choose n_{1} } \cdot \ p^{ n_{1} } \cdot \ q^{ n-n_{1} }=252 \cdot \ 0.25^{ 5 } \cdot \ 0.75^{ 10-5 }=0.058
C10(10)=(1010)=10!10!(1010)!=11=1 b=(nn) pn qnn=1 0.2510 0.751010=0C_{{ 10}}(10) = \dbinom{ 10}{ 10} = \dfrac{ 10! }{ 10!(10-10)!} = \dfrac{ 1 } { 1 } = 1 \ \\ b={ { n } \choose n } \cdot \ p^{ n } \cdot \ q^{ n-n }=1 \cdot \ 0.25^{ 10 } \cdot \ 0.75^{ 10-10 }=0
C0(10)=(100)=10!0!(100)!=11=1 c=(n0) p0 qn0=1 0.250 0.75100=0.056C_{{ 0}}(10) = \dbinom{ 10}{ 0} = \dfrac{ 10! }{ 0!(10-0)!} = \dfrac{ 1 } { 1 } = 1 \ \\ c={ { n } \choose 0 } \cdot \ p^{ 0 } \cdot \ q^{ n-0 }=1 \cdot \ 0.25^{ 0 } \cdot \ 0.75^{ 10-0 }=0.056



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