Tugas Pertemuan 12
Studi Kasus 1
Diketahui
Probabilitas:
Probabilitas produk cacat \(P(D = \text{Yes}) = 5\% = 0.05\)
Probabilitas produk tidak cacat \(P(D = \text{No}) = 95\% = 0.95\)
Probabilitas penggunaan komponen berkualitas rendah \(P(C = \text{Low} | D = \text{Yes}) = 80\% = 0.8\)
Probabilitas penggunaan komponen berkualitas rendah \(P(C = \text{Low} | D = \text{No}) = 20\% = 0.2\)
Probabilitas proses produksi di bawah standar \(P(P = \text{Below} | D = \text{Yes}) = 70\% = 0.7\)
Probabilitas proses produksi di bawah standar \(P(P = \text{Below} | D = \text{No}) = 30\% = 0.3\)
Pertanyaan Berapa probabilitas bahwa suatu produk akan cacat (\(D = \text{Yes}\)), jika diketahui:
Komponen yang digunakan berkualitas rendah (\(C = \text{Low}\)),
Proses produksi dilakukan di bawah standar (\(P = \text{Below}\)).
Gunakan Teorema Bayes.
Jawaban
Gunakan formula Teorema Bayes:
\[ P(D = \text{Yes} | C = \text{Low}, P = \text{Below}) = \frac{P(D = \text{Yes}) \cdot P(C = \text{Low} | D = \text{Yes}) \cdot P(P = \text{Below} | D = \text{Yes})}{P(C = \text{Low}, P = \text{Below})} \]
Langkah-langkah
1. Hitung \(P(D = \text{Yes}, C = \text{Low}, P = \text{Below})\):
Rumus: \[ P(D = \text{Yes}, C = \text{Low}, P = \text{Below}) = P(D = \text{Yes}) \cdot P(C = \text{Low} | D = \text{Yes}) \cdot P(P = \text{Below} | D = \text{Yes}) \]
Substitusi angka: \[ P(D = \text{Yes}, C = \text{Low}, P = \text{Below}) = 0.05 \cdot 0.8 \cdot 0.7 \]
Hasil: \[ P(D = \text{Yes}, C = \text{Low}, P = \text{Below}) = 0.028 \]
2. Hitung \(P(C = \text{Low}, P = \text{Below})\):
Menggunakan aturan total probabilitas: \[ P(C = \text{Low}, P = \text{Below}) = P(D = \text{Yes}) \cdot P(C = \text{Low}, P = \text{Below} | D = \text{Yes}) + P(D = \text{No}) \cdot P(C = \text{Low}, P = \text{Below} | D = \text{No}) \]
Bagian 1: \(P(D = \text{Yes}) \cdot P(C = \text{Low}, P = \text{Below} | D = \text{Yes})\):
Rumus: \[ P(D = \text{Yes}) \cdot P(C = \text{Low}, P = \text{Below} | D = \text{Yes}) = P(D = \text{Yes}) \cdot P(C = \text{Low} | D = \text{Yes}) \cdot P(P = \text{Below} | D = \text{Yes}) \]
Substitusi angka: \[ P(D = \text{Yes}) \cdot P(C = \text{Low}, P = \text{Below} | D = \text{Yes}) = 0.05 \cdot 0.8 \cdot 0.7 \]
Hasil: \[ P(D = \text{Yes}) \cdot P(C = \text{Low}, P = \text{Below} | D = \text{Yes}) = 0.028 \]
Bagian 2: \(P(D = \text{No}) \cdot P(C = \text{Low}, P = \text{Below} | D = \text{No})\):
Rumus: \[ P(D = \text{No}) \cdot P(C = \text{Low}, P = \text{Below} | D = \text{No}) = P(D = \text{No}) \cdot P(C = \text{Low} | D = \text{No}) \cdot P(P = \text{Below} | D = \text{No}) \]
Substitusi angka: \[ P(D = \text{No}) \cdot P(C = \text{Low}, P = \text{Below} | D = \text{No}) = 0.95 \cdot 0.2 \cdot 0.3 \]
Hasil: \[ P(D = \text{No}) \cdot P(C = \text{Low}, P = \text{Below} | D = \text{No}) = 0.057 \]
Total \(P(C = \text{Low}, P = \text{Below})\): \[ P(C = \text{Low}, P = \text{Below}) = 0.028 + 0.057 = 0.085 \]
3. Hitung \(P(D = \text{Yes} | C = \text{Low}, P = \text{Below})\):
Rumus: \[ P(D = \text{Yes} | C = \text{Low}, P = \text{Below}) = \frac{P(D = \text{Yes}, C = \text{Low}, P = \text{Below})}{P(C = \text{Low}, P = \text{Below})} \]
Substitusi angka: \[ P(D = \text{Yes} | C = \text{Low}, P = \text{Below}) = \frac{0.028}{0.085} \]
Hasil: \[ P(D = \text{Yes} | C = \text{Low}, P = \text{Below}) = 0.3294 \]
Hasil Akhir
Probabilitas bahwa suatu produk akan cacat (\(D = \text{Yes}\)), jika diketahui bahwa produk menggunakan komponen berkualitas rendah (\(C = \text{Low}\)) dan proses produksinya di bawah standar (\(P = \text{Below}\)), adalah 32.94% atau 0.3294.
Studi Kasus 2
Diketahui
Probabilitas dasar:
\(P(F) = 0.01\) (Probabilitas transaksi adalah penipuan).
\(P(\neg F) = 0.99\) (Probabilitas transaksi bukan penipuan).
Kombinasi karakteristik:
Lokasi asing: \(P(L = \text{Foreign}) = 0.20\),
Jumlah transaksi tinggi: \(P(A = \text{High}) = 0.10\),
Metode pembayaran kartu kredit: \(P(M = \text{CreditCard}) = 0.50\).
Probabilitas bersyarat untuk transaksi penipuan:
\(P(L = \text{Foreign} | F) = 0.40\),
\(P(A = \text{High} | F) = 0.50\),
\(P(M = \text{CreditCard} | F) = 0.60\).
Probabilitas bersyarat untuk transaksi non-penipuan:
\(P(L = \text{Foreign} | \neg F) = 0.10\),
\(P(A = \text{High} | \neg F) = 0.05\),
\(P(M = \text{CreditCard} | \neg F) = 0.30\).
Langkah Langkah
2. Hitung \(P(\text{Combination} | F)\)
Probabilitas semua kondisi terjadi bersamaan, diberikan transaksi adalah penipuan: \[ P(\text{Combination} | F) = P(L = \text{Foreign} | F) \cdot P(A = \text{High} | F) \cdot P(M = \text{CreditCard} | F) \]
Substitusi angka: \[ P(\text{Combination} | F) = 0.40 \cdot 0.50 \cdot 0.60 \]
Perhitungan: \[ P(\text{Combination} | F) = 0.12 \]
3. Hitung \(P(\text{Combination} | \neg F)\)
Probabilitas semua kondisi terjadi bersamaan, diberikan transaksi bukan penipuan: \[ P(\text{Combination} | \neg F) = P(L = \text{Foreign} | \neg F) \cdot P(A = \text{High} | \neg F) \cdot P(M = \text{CreditCard} | \neg F) \]
Substitusi angka: \[ P(\text{Combination} | \neg F) = 0.10 \cdot 0.05 \cdot 0.30 \]
Perhitungan: \[ P(\text{Combination} | \neg F) = 0.0015 \]
4. Hitung \(P(\text{Combination})\)
Menggunakan aturan total probabilitas: \[ P(\text{Combination}) = P(\text{Combination} | F) \cdot P(F) + P(\text{Combination} | \neg F) \cdot P(\neg F) \]
Substitusi angka: \[ P(\text{Combination}) = (0.12 \cdot 0.01) + (0.0015 \cdot 0.99) \]
Perhitungan: \[ P(\text{Combination}) = 0.0012 + 0.001485 = 0.002685 \]
5. Hitung \(P(F | \text{Combination})\)
Gunakan Teorema Bayes: \[ P(F | \text{Combination}) = \frac{P(\text{Combination} | F) \cdot P(F)}{P(\text{Combination})} \]
Substitusi angka: \[ P(F | \text{Combination}) = \frac{0.12 \cdot 0.01}{0.002685} \]
Perhitungan: \[ P(F | \text{Combination}) = \frac{0.0012}{0.002685} = 0.4472 \]
Konversi ke persentase: \[ P(F | \text{Combination}) = 44.72\% \]
Hasil Akhir
Probabilitas transaksi adalah penipuan, diberikan kombinasi karakteristik tersebut, adalah 44.72%.