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Ali mendapat tugas mengerjakan soal pilihan ganda sebanyak 8 soal dengan 5 opsi jawaban A, B, C, D, dan E. Ali hanya menebak jawaban setiap soal. Tentukan peluang Ali menjawab benar paling banyak: a. 2 soal b. 3 soal c. 4 soal

Pertanyaan

Ali mendapat tugas mengerjakan soal pilihan ganda sebanyak 8 soal dengan 5 opsi jawaban A, B, C, D, dan E. Ali hanya menebak jawaban setiap soal. Tentukan peluang Ali menjawab benar paling banyak:

a. 2 soal

b. 3 soal

c. 4 soal

D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Jawaban

peluang paling banyak 4 soal yang benar adalah 0 comma 958.

Pembahasan

a. Jawaban yang benar untuk pertanyaan tersebut adalah bold 0 bold comma bold 794.

Permasalahan tersebut berkaitan dengan fungsi distribusi binomial kumulatif atau straight P left parenthesis X less or equal than t right parenthesis equals sum from x equals 0 to t of straight C left parenthesis n comma space x right parenthesis times p to the power of x times q to the power of n minus x end exponent.

Ingat kembali rumus kombinasi straight C italic left parenthesis n italic comma italic space x right parenthesis equals fraction numerator n factorial over denominator left parenthesis n minus x right parenthesis factorial times x factorial end fraction.

Pada soal diketahui:

Banyak soal yang dikerjakan left parenthesis n right parenthesis equals 8.

Peluang jawaban yang dipilih benar left parenthesis p right parenthesis equals 1 fifth equals 0 comma 2.

Peluang jawaban yang dipilih salah left parenthesis q right parenthesis equals 1 minus p equals 4 over 5 equals 0 comma 8.

Karena yang ditanyakan adalah peluang paling banyak 2 soal yang benar, maka x equals 0 comma space 1 comma space 2 sehingga straight P left parenthesis X less or equal than 2 right parenthesis equals straight P left parenthesis 0 right parenthesis plus straight P left parenthesis 1 right parenthesis plus straight P left parenthesis 2 right parenthesis

table attributes columnalign right center left columnspacing 0px end attributes row cell straight P left parenthesis 0 right parenthesis end cell equals cell straight C left parenthesis 8 comma space 0 right parenthesis times left parenthesis 0 comma 2 right parenthesis to the power of 0 times left parenthesis 0 comma 8 right parenthesis to the power of 8 minus 0 end exponent space end cell row blank equals cell fraction numerator 8 factorial over denominator left parenthesis 8 minus 0 right parenthesis factorial times 0 factorial end fraction left parenthesis 1 right parenthesis left parenthesis 0 comma 8 right parenthesis to the power of 8 space end cell row blank equals cell 1 left parenthesis 1 right parenthesis left parenthesis 0 comma 167 right parenthesis equals 0 comma 167 end cell row cell straight P left parenthesis 1 right parenthesis end cell equals cell straight C left parenthesis 8 comma space 1 right parenthesis times left parenthesis 0 comma 2 right parenthesis to the power of 1 left parenthesis 0 comma 8 right parenthesis to the power of 8 minus 1 end exponent space space end cell row blank equals cell fraction numerator 8 factorial over denominator left parenthesis 8 minus 1 right parenthesis factorial times 1 factorial end fraction left parenthesis 0 comma 2 right parenthesis left parenthesis 0 comma 8 right parenthesis to the power of 7 end cell row blank equals cell fraction numerator 8 cross times up diagonal strike 7 factorial end strike over denominator up diagonal strike 7 factorial end strike left parenthesis 1 right parenthesis end fraction left parenthesis 0 comma 2 right parenthesis left parenthesis 0 comma 209 right parenthesis equals 0 comma 334 end cell row cell straight P left parenthesis 2 right parenthesis end cell equals cell straight C left parenthesis 8 comma space 2 right parenthesis times left parenthesis 0 comma 2 right parenthesis squared left parenthesis 0 comma 8 right parenthesis to the power of 8 minus 2 end exponent space space end cell row blank equals cell fraction numerator 8 factorial over denominator left parenthesis 8 minus 2 right parenthesis factorial times 2 factorial end fraction left parenthesis 0 comma 04 right parenthesis left parenthesis 0 comma 8 right parenthesis to the power of 6 end cell row blank equals cell fraction numerator scriptbase up diagonal strike 8 end scriptbase presuperscript 4 cross times 7 cross times up diagonal strike 6 factorial end strike over denominator up diagonal strike 6 factorial times end strike up diagonal strike left parenthesis 2 right parenthesis end strike end fraction left parenthesis 0 comma 04 right parenthesis left parenthesis 0 comma 262 right parenthesis equals 0 comma 293 end cell end table 

Kumulatifkan  straight P left parenthesis X less or equal than 2 right parenthesis equals straight P left parenthesis 0 right parenthesis plus straight P left parenthesis 1 right parenthesis plus straight P left parenthesis 2 right parenthesis.

table attributes columnalign right center left columnspacing 0px end attributes row cell straight P left parenthesis X end cell less or equal than cell 2 right parenthesis equals straight P left parenthesis 0 right parenthesis plus straight P left parenthesis 1 right parenthesis plus straight P left parenthesis 2 right parenthesis end cell row blank blank cell equals 0 comma 167 plus 0 comma 334 plus 0 comma 293 equals 0 comma 794 end cell end table 

Dengan demikian, peluang paling banyak 2 soal yang benar adalah 0 comma 794.

b. Jawaban yang benar untuk pertanyaan tersebut adalah bold 0 bold comma bold 94.

Masih berkaitan dengan fungsi distribusi binomial kumulatif, namun karena yang ditanyakan adalah peluang paling banyak benar 3 soal maka x equals 0 comma space 1 comma space 2 comma space 3 sehingga straight P left parenthesis X less or equal than 3 right parenthesis equals straight P left parenthesis 0 right parenthesis plus straight P left parenthesis 1 right parenthesis plus straight P left parenthesis 2 right parenthesis plus straight P left parenthesis 3 right parenthesis.

Tentukan straight P left parenthesis 3 right parenthesis.

table attributes columnalign right center left columnspacing 0px end attributes row cell straight P left parenthesis 3 right parenthesis end cell equals cell straight C left parenthesis 8 comma space 3 right parenthesis times left parenthesis 0 comma 2 right parenthesis cubed times left parenthesis 0 comma 8 right parenthesis to the power of 8 minus 3 end exponent space end cell row blank equals cell fraction numerator 8 factorial over denominator left parenthesis 8 minus 3 right parenthesis factorial times 3 factorial end fraction left parenthesis 0 comma 008 right parenthesis left parenthesis 0 comma 8 right parenthesis to the power of 5 end cell row blank equals cell fraction numerator 8 cross times 7 cross times up diagonal strike 6 cross times up diagonal strike 5 factorial over denominator up diagonal strike 5 factorial cross times end strike up diagonal strike left parenthesis 6 right parenthesis end strike end fraction left parenthesis 0 comma 008 right parenthesis left parenthesis 0 comma 327 right parenthesis equals 0 comma 146 end cell end table 

Dengan melihat kembali nilai straight P left parenthesis 0 right parenthesis comma space straight P left parenthesis 1 right parenthesis comma space straight P left parenthesis 2 right parenthesis, kumulatifkan straight P left parenthesis X less or equal than 3 right parenthesis equals straight P left parenthesis 0 right parenthesis plus straight P left parenthesis 1 right parenthesis plus straight P left parenthesis 2 right parenthesis plus straight P left parenthesis 3 right parenthesis.

table attributes columnalign right center left columnspacing 0px end attributes row cell straight P left parenthesis X end cell less or equal than cell 3 right parenthesis equals straight P left parenthesis 0 right parenthesis plus straight P left parenthesis 1 right parenthesis plus straight P left parenthesis 2 right parenthesis plus straight P left parenthesis 3 right parenthesis end cell row blank blank cell equals 0 comma 167 plus 0 comma 334 plus 0 comma 293 plus 0 comma 146 equals 0 comma 94 end cell end table 

Dengan demikian, peluang paling banyak 3 soal yang benar adalah 0 comma 94.

c. Jawaban yang benar untuk pertanyaan tersebut adalah bold 0 bold comma bold 985.

Masih berkaitan dengan fungsi distribusi binomial kumulatif, namun karena yang ditanyakan adalah peluang paling banyak benar 4 soal maka x equals 0 comma space 1 comma space 2 comma space 3 comma space 4 sehingga straight P left parenthesis X less or equal than 4 right parenthesis equals straight P left parenthesis 0 right parenthesis plus straight P left parenthesis 1 right parenthesis plus straight P left parenthesis 2 right parenthesis plus straight P left parenthesis 3 right parenthesis plus straight P left parenthesis 4 right parenthesis.

Tentukan straight P left parenthesis 4 right parenthesis.

table attributes columnalign right center left columnspacing 0px end attributes row cell straight P left parenthesis 4 right parenthesis end cell equals cell straight C left parenthesis 8 comma space 4 right parenthesis times left parenthesis 0 comma 2 right parenthesis to the power of 4 times left parenthesis 0 comma 8 right parenthesis to the power of 8 minus 4 end exponent space end cell row blank equals cell fraction numerator 8 factorial over denominator left parenthesis 8 minus 4 right parenthesis factorial times 4 factorial end fraction left parenthesis 0 comma 0016 right parenthesis left parenthesis 0 comma 8 right parenthesis to the power of 4 end cell row blank equals cell fraction numerator scriptbase up diagonal strike 8 end scriptbase presuperscript 2 cross times 7 cross times up diagonal strike 6 cross times 5 cross times up diagonal strike 4 factorial over denominator up diagonal strike 4 factorial cross times end strike up diagonal strike 4 cross times up diagonal strike 3 cross times 2 cross times 1 end strike end fraction left parenthesis 0 comma 0016 right parenthesis left parenthesis 0 comma 4096 right parenthesis equals 0 comma 0458 end cell end table  

Dengan melihat kembali nilai straight P left parenthesis 0 right parenthesis comma space straight P left parenthesis 1 right parenthesis comma space straight P left parenthesis 2 right parenthesis comma space straight P left parenthesis 3 right parenthesis, kumulatifkan straight P left parenthesis X less or equal than 4 right parenthesis equals straight P left parenthesis 0 right parenthesis plus straight P left parenthesis 1 right parenthesis plus straight P left parenthesis 2 right parenthesis plus straight P left parenthesis 3 right parenthesis plus straight P left parenthesis 4 right parenthesis.

table attributes columnalign right center left columnspacing 0px end attributes row cell straight P left parenthesis X end cell less or equal than cell 4 right parenthesis equals straight P left parenthesis 0 right parenthesis plus straight P left parenthesis 1 right parenthesis plus straight P left parenthesis 2 right parenthesis plus straight P left parenthesis 3 right parenthesis plus straight P left parenthesis 4 right parenthesis end cell row blank blank cell equals 0 comma 167 plus 0 comma 334 plus 0 comma 293 plus 0 comma 146 plus 0 comma 0458 equals 0 comma 985 end cell end table  

Dengan demikian, peluang paling banyak 4 soal yang benar adalah 0 comma 958.

2rb+

5.0 (3 rating)

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