Pembahasan Soal Ujian Profesi Aktuaris
Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
Mata Ujian | : | Probabilitas dan Statistika |
Periode Ujian | : | November 2018 |
Nomor Soal | : | 6 |
SOAL
Misal A, B, C, dan D memiliki kejadian sebagai berikut:
\(B = A’\)\(;C \cap D = \Phi \)
\(P[A] = \frac{1}{4};P[B] = \frac{3}{4}\,\)
\(P[C|A] = \frac{1}{2};P[C|B] = \frac{3}{4};P[D|A] = \frac{1}{4};P[D|B] = \frac{1}{8}\)
Hitunglah \(P[C \cup D]\)
- \(\frac{5}{{32}}\)
- \(\frac{1}{4}\)
- \(\frac{{27}}{{32}}\)
- \(\frac{3}{4}\)
- 1
Step 1 | | C | C’ | | A | 1/8 | 1/8 | 1/4 | A’ | 9/16 | 3/16 | 3/4 | | 11/16 | 5/16 | 1 | | D | D’ | | B’ | 1/16 | 3/16 | 1/4 | B | 3/32 | 21/32 | 3/4 | | 5/32 | 27/32 | 1 |
\(P[C|A] = \frac{{P[C \cap A]}}{{P[A]}}\)
\(P[C \cap A] = \frac{1}{2}\,\frac{1}{4}\,\, = \,\frac{1}{8}\)
\(P[C|B] = \frac{{P[C \cap B]}}{{P[B]}}\)
\(P[C \cap B] = \frac{3}{4}\,\frac{3}{4}\,\, = \,\frac{9}{{16}}\)
\(P[D|A] = \frac{{P[D \cap A]}}{{P[A]}}\)
\(P[D \cap A] = \frac{1}{4}\,\frac{1}{4}\,\, = \,\frac{1}{{16}}\)
\(P[D|B] = \frac{{P[D \cap B]}}{{P[B]}}\)
\(P[D \cap B] = \frac{1}{8}\,\frac{3}{4}\,\, = \,\frac{3}{{32}}\) |
Step 2 |
\(P[C \cup D] = \frac{{11}}{{16}} + \frac{5}{{32}} – 0\)
\(P[C \cup D] = \frac{{27}}{{32}}\) |
Jawaban | d. \(\frac{{27}}{{32}}\) |