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. \(\frac{{27}}{{32}}\) |