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Tujuh uang logam seimbang dilantunkan. Tentukan probabilitas: a. munculnya sisi gambar tepat lima kali. b.munculnya sisi gambar sedikitnya 4 kali. c. munculnya sisi gambar paling banyak 2 kali.

Pertanyaan

Tujuh uang logam seimbang dilantunkan. Tentukan probabilitas:

a. munculnya sisi gambar tepat lima kali.

b.munculnya sisi gambar sedikitnya 4 kali.

c. munculnya sisi gambar paling banyak 2 kali.

A. Salim

Master Teacher

Mahasiswa/Alumni Universitas Pelita Harapan

Jawaban terverifikasi

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah a. Probabilitas munculnya gambar sebanyak 5  kali adalah 0 comma 164, b. probabiltas munculnya sisi gambar sedikitnya  4 kali adalah 0 comma 5, dan c.probabilitas munculnya sisi gambar paling banyak 2 kali adalah 0 comma 227

Ingat!

Rumus Distribusi Binomial left parenthesis P right parenthesis

P left parenthesis x comma n right parenthesis equals C left parenthesis n comma x right parenthesis times p to the power of x times q to the power of n minus x end exponent.

Dimana, :p Prorabilitas Sukses, dan q: Probabilitas Gagal.

Sehingga, 

Pada  percobaan 7 koin dilantunkan, maka n equals 7, karena koin seimbang maka peluang gambar p equals 1 half, dan peluang bukan gambar q equals 1 half. maka

a. munculnya sisi gambar tepat 5 kali maka x equals 5.

Jadi

table attributes columnalign right center left columnspacing 0px end attributes row cell P left parenthesis x comma n right parenthesis end cell equals cell C left parenthesis n comma x right parenthesis times p to the power of x times q to the power of n minus x end exponent end cell row cell P left parenthesis 5 comma 7 right parenthesis end cell equals cell C left parenthesis 7 comma 5 right parenthesis times open parentheses 1 half close parentheses to the power of 5 times open parentheses 1 half close parentheses to the power of 7 minus 5 end exponent end cell row blank equals cell fraction numerator 7 factorial over denominator 5 factorial times open parentheses 7 minus 5 close parentheses factorial end fraction times open parentheses 1 half close parentheses to the power of 5 times open parentheses 1 half close parentheses squared end cell row blank equals cell fraction numerator 7 cross times 6 cross times 5 factorial over denominator 5 factorial cross times 2 factorial end fraction times open parentheses 1 over 32 close parentheses times open parentheses 1 fourth close parentheses end cell row blank equals cell 42 over 2 times 1 over 32 times 1 fourth end cell row blank equals cell 21 over 128 end cell row blank equals cell 0 comma 164 end cell end table

Dengan demikian, didapat Probabilitas munculnya gambar sebanyak 5 kali adalah 0 comma 164.

b. munculnya sisi gambar sedikitnya  4 kali maka x greater or equal than 4.

Jadi

P left parenthesis x greater or equal than 4 right parenthesis equals P left parenthesis x equals 4 right parenthesis plus P left parenthesis x equals 5 right parenthesis plus P left parenthesis x equals 6 right parenthesis plus P left parenthesis x equals 7 right parenthesis

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell table attributes columnalign center center left end attributes row cell P left parenthesis x equals 4 right parenthesis end cell equals cell C left parenthesis 7 comma 4 right parenthesis times open parentheses 1 half close parentheses to the power of 4 times open parentheses 1 half close parentheses to the power of 7 minus 4 end exponent end cell row space equals cell fraction numerator 7 factorial over denominator 4 factorial times left parenthesis 7 minus 4 right parenthesis factorial end fraction times open parentheses 1 half close parentheses to the power of 4 times open parentheses 1 half close parentheses cubed end cell row space equals cell fraction numerator 7 cross times 6 cross times 5 cross times 4 factorial over denominator 4 factorial times 3 factorial end fraction times 1 over 16 times 1 over 8 end cell row space equals cell 210 over 6 times 1 over 128 end cell row space equals cell 105 over 384 end cell row space equals cell 0 comma 273 end cell end table end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell table attributes columnalign center center left end attributes row cell P left parenthesis x equals 5 right parenthesis end cell equals cell C left parenthesis 7 comma 5 right parenthesis times open parentheses 1 half close parentheses to the power of 5 times open parentheses 1 half close parentheses to the power of 7 minus 5 end exponent end cell row space equals cell fraction numerator 7 factorial over denominator 5 factorial times left parenthesis 7 minus 5 right parenthesis factorial end fraction times open parentheses 1 half close parentheses to the power of 5 times open parentheses 1 half close parentheses squared end cell row space equals cell fraction numerator 7 cross times 6 cross times 5 factorial over denominator 5 factorial times 2 factorial end fraction times 1 over 32 times 1 fourth end cell row space equals cell 42 over 2 times 1 over 128 end cell row space equals cell 21 over 128 end cell row space equals cell 0 comma 164 end cell end table end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell table attributes columnalign center center left end attributes row cell P left parenthesis x equals 6 right parenthesis end cell equals cell C left parenthesis 7 comma 6 right parenthesis times open parentheses 1 half close parentheses to the power of 6 times open parentheses 1 half close parentheses to the power of 7 minus 6 end exponent end cell row space equals cell fraction numerator 7 factorial over denominator 6 factorial times left parenthesis 7 minus 6 right parenthesis factorial end fraction times open parentheses 1 half close parentheses to the power of 6 times open parentheses 1 half close parentheses to the power of 1 end cell row space equals cell fraction numerator 7 cross times 6 factorial over denominator 6 factorial times 1 factorial end fraction times 1 over 64 times 1 half end cell row space equals cell 7 over 1 times 1 over 128 end cell row space equals cell 7 over 128 end cell row space equals cell 0 comma 055 end cell end table end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell table attributes columnalign center center left end attributes row cell P left parenthesis x equals 7 right parenthesis end cell equals cell C left parenthesis 7 comma 7 right parenthesis times open parentheses 1 half close parentheses to the power of 7 times open parentheses 1 half close parentheses to the power of 7 minus 7 end exponent end cell row space equals cell fraction numerator 7 factorial over denominator 7 factorial times left parenthesis 7 minus 7 right parenthesis factorial end fraction times open parentheses 1 half close parentheses to the power of 7 times open parentheses 1 half close parentheses to the power of 0 end cell row space equals cell fraction numerator 7 factorial over denominator 7 factorial times 0 factorial end fraction times 1 over 128 times 1 end cell row space equals cell 1 times 1 over 128 end cell row space equals cell 1 over 128 end cell row space equals cell 0 comma 008 end cell end table end cell end table

Dengan demikian, didapat probabiltas munculnya sisi gambar sedikitnya  4 kali adalah

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell table attributes columnalign center center left end attributes row cell P left parenthesis x greater or equal than 4 right parenthesis end cell equals cell P left parenthesis x equals 4 right parenthesis plus P left parenthesis x equals 5 right parenthesis plus P left parenthesis x equals 6 right parenthesis plus P left parenthesis x equals 7 right parenthesis end cell row space equals cell open parentheses 0 comma 273 close parentheses plus open parentheses 0 comma 164 close parentheses plus left parenthesis 0 comma 055 right parenthesis plus open parentheses 0 comma 008 close parentheses end cell row blank equals cell 0 comma 5 end cell end table end cell end table

c. munculnya sisi gambar paling banyak 2 kali. left parenthesis x less or equal than 2 right parenthesis

P left parenthesis x less or equal than 2 right parenthesis equals P left parenthesis x equals 0 right parenthesis plus P left parenthesis x equals 1 right parenthesis plus P left parenthesis x equals 2 right parenthesis

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell table attributes columnalign center center left end attributes row cell P left parenthesis x equals 0 right parenthesis end cell equals cell C left parenthesis 7 comma 0 right parenthesis times open parentheses 1 half close parentheses to the power of 0 times open parentheses 1 half close parentheses to the power of 7 minus 0 end exponent end cell row space equals cell fraction numerator 7 factorial over denominator 0 factorial times left parenthesis 7 minus 0 right parenthesis factorial end fraction times open parentheses 1 half close parentheses to the power of 0 times open parentheses 1 half close parentheses to the power of 7 end cell row space equals cell fraction numerator 7 factorial over denominator 0 factorial times 7 factorial end fraction times 1 times 1 over 128 end cell row space equals cell 1 times 1 over 128 end cell row space equals cell 1 over 128 end cell row space equals cell 0 comma 008 end cell end table end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell table attributes columnalign center center left end attributes row cell P left parenthesis x equals 1 right parenthesis end cell equals cell C left parenthesis 7 comma 1 right parenthesis times open parentheses 1 half close parentheses to the power of 1 times open parentheses 1 half close parentheses to the power of 7 minus 1 end exponent end cell row space equals cell fraction numerator 7 factorial over denominator 1 factorial times left parenthesis 7 minus 1 right parenthesis factorial end fraction times open parentheses 1 half close parentheses to the power of 1 times open parentheses 1 half close parentheses to the power of 6 end cell row space equals cell fraction numerator 7 cross times 6 factorial over denominator 1 factorial times 6 factorial end fraction times 1 half times 1 over 64 end cell row space equals cell 7 over 1 times 1 over 128 end cell row space equals cell 7 over 128 end cell row space equals cell 0 comma 055 end cell end table end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell table attributes columnalign center center left end attributes row cell P left parenthesis x equals 2 right parenthesis end cell equals cell C left parenthesis 7 comma 2 right parenthesis times open parentheses 1 half close parentheses squared times open parentheses 1 half close parentheses to the power of 7 minus 2 end exponent end cell row space equals cell fraction numerator 7 factorial over denominator 2 factorial times left parenthesis 7 minus 2 right parenthesis factorial end fraction times open parentheses 1 half close parentheses squared times open parentheses 1 half close parentheses to the power of 5 end cell row space equals cell fraction numerator 7 cross times 6 cross times 5 factorial over denominator 2 factorial times 5 factorial end fraction times 1 fourth times 1 over 32 end cell row space equals cell 42 over 2 times 1 over 128 end cell row space equals cell 21 over 128 end cell row space equals cell 0 comma 164 end cell end table end cell end table

Dengan demikian, didapat probabilitas munculnya sisi gambar paling banyak 2 kali adalah

table attributes columnalign center center left end attributes row cell P left parenthesis x less or equal than 2 right parenthesis end cell equals cell P left parenthesis x equals 0 right parenthesis plus P left parenthesis x equals 1 right parenthesis plus P left parenthesis x equals 2 right parenthesis end cell row blank equals cell open parentheses 0 comma 008 close parentheses plus open parentheses 0 comma 055 close parentheses plus open parentheses 0 comma 164 close parentheses end cell row blank equals cell 0 comma 227 end cell end table

325

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