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 $0,164$ , b. probabiltas munculnya sisi gambar sedikitnya $4$ kali adalah $0,5$ , dan c.probabilitas munculnya sisi gambar paling banyak $2$ kali adalah $0,227$ .

Ingat!

Rumus Distribusi Binomial $(P)$

$P(x,n)=C(n,x)\cdot p^x\cdot q^{n-x}$ .

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

Sehingga,

Pada percobaan $7$ koin dilantunkan, maka $n=7$ , karena koin seimbang maka peluang gambar $p=\frac12$ , dan peluang bukan gambar $q=\frac12$ . maka

a. munculnya sisi gambar tepat $5$ kali maka $x=5$ .

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 $0,164$ .

b. munculnya sisi gambar sedikitnya $4$ kali maka $x\geq4$ .

Jadi

$P(x\geq4)=P(x=4)+P(x=5)+P(x=6)+P(x=7)$

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

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

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

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

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

$\begin{array}{rcl}&&\begin{array}{ccl}{P(x}\geq{4)}&=&{P(x}={4)+P(x=5)+P(x=6)+P(x=7)}\\\;&=&\left(0,273\right)+\left(0,164\right)+(0,055)+\left(0,008\right)\\&=&0,5\end{array}\end{array}$

c. munculnya sisi gambar paling banyak $2$ kali. $(x\leq2)$

$P(x\leq2)=P(x=0)+P(x=1)+P(x=2)$

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

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

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

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

$\begin{array}{ccl}P(x\leq2)&=&P(x=0)+P(x=1)+P(x=2)\\&=&\left(0,008\right)+\left(0,055\right)+\left(0,164\right)\\&=&0,227\end{array}$

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