Konsep Dasar Probabilitas

1 Penerapan Probabilitas dalam Prediksi Kualitas Produk

1.1 Diketahui:

  1. Probabilitas produk cacat:

    • \(P(D = \text{Yes}) = 5\% = 0.05\)

  2. Probabilitas produk tidak cacat:

    • \(P(D = \text{No}) = 95\% = 0.95\)

  3. Probabilitas menggunakan komponen berkualitas rendah:

    • \(P(C = \text{Low}) = 30\% = 0.3\)

  4. Probabilitas proses produksi di bawah standar:

    • \(P(P = \text{Below}) = 40\% = 0.4\)

  5. Probabilitas bersama:

    • \(P(C = \text{Low} \cap P = \text{Below} \mid D = \text{Yes}) = 0.02\)

    • \(P(C = \text{Low} \cap P = \text{Below} \mid D = \text{No}) = 0.28\)

1.2 Langkah 1: Rumus Teorema Bayes

Menggunakan Teorema Bayes, probabilitas bersyarat dihitung sebagai:

\[ P(D = \text{Yes} \mid C = \text{Low}, P = \text{Below}) = \frac{P(C = \text{Low}, P = \text{Below} \mid D = \text{Yes}) \cdot P(D = \text{Yes})}{P(C = \text{Low}, P = \text{Below})} \]

1.3 Langkah 2: Hitung Denominator \(P(C = \text{Low}, P = \text{Below})\)

Menggunakan aturan probabilitas total:

\[ P(C = \text{Low}, P = \text{Below}) = P(C = \text{Low}, P = \text{Below} \mid D = \text{Yes}) \cdot P(D = \text{Yes}) + P(C = \text{Low}, P = \text{Below} \mid D = \text{No}) \cdot P(D = \text{No}) \]

Substitusi nilai:

\[ P(C = \text{Low}, P = \text{Below}) = (0.02 \cdot 0.05) + (0.28 \cdot 0.95) \] \[ P(C = \text{Low}, P = \text{Below}) = 0.001 + 0.266 = 0.267 \]

1.4 Langkah 3: Hitung Probabilitas Bersyarat

Substitusi nilai ke dalam rumus:

\[ P(D = \text{Yes} \mid C = \text{Low}, P = \text{Below}) = \frac{(0.02 \cdot 0.05)}{0.267} \] \[ P(D = \text{Yes} \mid C = \text{Low}, P = \text{Below}) = \frac{0.001}{0.267} \approx 0.0037 \]

1.5 Hasil Akhir:

Probabilitas bahwa suatu produk akan cacat jika diketahui komponen berkualitas rendah dan proses produksi di bawah standar adalah 0.37%.

1.6 Menghitung Ukuran Sampel

## Ukuran sampel yang diperlukan:  567

2 Menghitung Probabilitas Penipuan Transaksi

Untuk menghitung probabilitas bahwa transaksi adalah penipuan dengan menggunakan Teorema Bayes, kita akan mengikuti langkah-langkah berikut:

2.1 Langkah 1: Menentukan Probabilitas Priori

  • \(P(F = \text{Fraud}) = 0.01\) (Probabilitas transaksi adalah penipuan)
  • \(P(F = \text{Not Fraud}) = 0.99\) (Probabilitas transaksi bukan penipuan)

2.2 Langkah 2: Menentukan Probabilitas Bersyarat

  • \(P(L = \text{Foreign}) = 0.20\) (Probabilitas transaksi dilakukan dari luar negeri)
  • \(P(A > 500) = 0.10\) (Probabilitas jumlah pembelian lebih dari $500)
  • \(P(M = \text{Credit Card}) = 0.50\) (Probabilitas menggunakan kartu kredit sebagai metode pembayaran)

2.3 Langkah 3: Menghitung Likelihood

Karena kita anggap fitur-fitur ini independen, kita dapat mengalikan probabilitas bersyarat untuk transaksi penipuan dan bukan penipuan.

  • Untuk transaksi penipuan \(F = \text{Fraud}\): \[ P(L, A, M \mid F = \text{Fraud}) = P(L = \text{Foreign}) \cdot P(A > 500) \cdot P(M = \text{Credit Card}) = 0.20 \times 0.10 \times 0.50 = 0.01 \]

  • Untuk transaksi bukan penipuan \(F = \text{Not Fraud}\): \[ P(L, A, M \mid F = \text{Not Fraud}) = P(L = \text{Foreign}) \cdot P(A > 500) \cdot P(M = \text{Credit Card}) = 0.20 \times 0.10 \times 0.50 = 0.01 \]

2.4 Langkah 4: Menghitung Probabilitas Total

Total probabilitas \(P(L, A, M)\) dihitung dengan menjumlahkan probabilitas untuk kedua kondisi (penipuan dan bukan penipuan):

\[ P(L, A, M) = P(L, A, M \mid F = \text{Fraud}) \cdot P(F = \text{Fraud}) + P(L, A, M \mid F = \text{Not Fraud}) \cdot P(F = \text{Not Fraud}) \] \[ P(L, A, M) = 0.01 \times 0.01 + 0.01 \times 0.99 = 0.01 \]

2.5 Langkah 5: Menghitung Probabilitas Bersyarat Menggunakan Teorema Bayes

Terakhir, kita dapat menghitung probabilitas bahwa transaksi adalah penipuan menggunakan rumus Teorema Bayes:

\[ P(F = \text{Fraud} \mid L = \text{Foreign}, A > 500, M = \text{Credit Card}) = \frac{P(L, A, M \mid F = \text{Fraud}) \cdot P(F = \text{Fraud})}{P(L, A, M)} \]

Substitusikan nilai-nilai yang sudah dihitung:

\[ P(F = \text{Fraud} \mid L = \text{Foreign}, A > 500, M = \text{Credit Card}) = \frac{0.01 \times 0.01}{0.01} = 0.01 \]

2.6 Hasil Akhir

Probabilitas transaksi tersebut adalah penipuan adalah 1% atau 0.01.

2.7 Menghitung Ukuran Sampel

## Ukuran sampel yang diperlukan:  385
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