WISKUNDE
GRAAD 10
NOG OEFENINGE
Waarskynlikheid : antwoorde.
ONTHOU die volgende:
MATHEMATICS
GRADE 10
MORE EXERCISES
Probability : answers.
REMEMBER the following :
P(A of B) = P(A) + P(B) − P(A en B)
of ook P(A ∪ B)= P(A) + P(B) − P(A ∩ B)
P(A en B) of ook P(A ∩ B)
P(nie A nie) = 1 − P(A)
of ook P(A') = 1 − P(A)
∅ stel die leê versameling voor.
P(A or B) = P(A) + P(B) − P(A and B)
or also P(A ∪ B)= P(A) + P(B) − P(A ∩ B)
P(A and B) or also P(A ∩ B)
P(not A) = 1 − P(A)
or also P(A') = 1 − P(A)
∅ represents the empty set.
15
1.1
P(A) = ───
1.2
P(nie / not B) = P(B') = 1 − P(B)
30
1
20
1
= ──
= 1 − ─── = ──
2
30
3
5
10
1.3
P(A en / and B) = P(A ∩ B) = ───
1.4
P(nie / not B en/and A) = P(B' ∩ A) = ───
30
30
1
1
= ──
= ───
6
3
10 + 5 + 15
15 − 5
1.5
P(A of / or B) = P(A ∪ B) = ────────
1.6
P(net / only A) = P(A) − P(A ∩ B) = ─────
30
30
1
= 1
= ───
3
30
2.2.1
2.1
P(N) = ───
50
3
= ──
5
22 + 10
2.2.2
P(nie/not H) = ─────
50
16
= ──
25
OF/OR P(nie/not H) = P(H') =1 − P(H)
18
16
= 1 − ─── = ──
50
25
8
10
2.2.3
P(N en/and H) = P(N ∩ H) = ───
2.2.4
P(nie/not N of/or H) = P((N ∪ H)') = ───
50
50
4
1
= ───
= ──
25
5
22
11
2.2.5
P(net / only N) = ─── = ───
50
25
2.3
Nee, N en H sny mekaar of hulle deel elemente of n(N ∩ H) ≠ 0 of N ∩ H ≠ ∅
No, N and H intersect or they share elements or n(N ∩ H) ≠ 0 or N ∩ H ≠ ∅
25
15
3.1
P(X) = ───
3.2
P(net/only Y) = ───
45
45
5
1
= ───
= ──
9
3
25
15
3.3
P(X) = ───
3.4
P(net/only Y) = ───
45
45
5
1
= ───
= ──
9
3
18 + 7 + 15
18 + 15
3.5
P(X of/orY) = ────────
3.6
P(net/only X of net/only Y) = ─────
45
45
8
11
= ───
= ──
9
15
4.1
n(A) = 6
4.2
n(B ∪ C) = 8
4.3
n((B ∩ C)') = 8
4.4
n(A') = 6
5
2
4.5
P(A) = ───
4.6
P((A ∪ B)') = ───
11
11
1
4.7
P(A) = ───
11
5.1
n(S) = 8 + 3 + 5 + 6 = 22
5.2
n(C) = 6 + 5 + 2 + 12 = 25
4
5.3
x + 2 + 5 + 3 + 8 + 6 + 5 = 45
5.4.1
P(net/only Tennis) = ───
45
x + 41 = 45
3
1
x = 4
5.4.2
P(net/only S en/and T) = ─── = ──
45
15
3
4 + 2 + 12 + 5
5.4.3
P(S ∩ T ∩ C) = ───
5.4.4
P(nie/not S) = ──────────
45
45
1
23
= ─
= ──
9
45
OF / OR P(nie/not S) = P(S') = 1 − P(S)
22
23
= 1 − ── = ──
45
45
6.1
A en B is onderling uitsluitend, sny mekaar nie.
6.2
A en C sny / A and C intersect
A and B are mutually exclusive, do not intersect.
∴ A ∩ B = ∅
∴ A ∩ C ≠ ∅
∴ n(A ∪ B) = n(A) + n(B)
n(A ∪ C) = n(A) + n(C) − n(A ∩ C)
25 = 15 + n(B)
25 = 11 + 18 − n(A ∩ C)
n(B) = 10
n(A ∩ C) = 4
6.3
n(A ∪ C) = n(A) + n(C) − n(A ∩ C)
49 = 22 + n(C) − 10
n(C) = 37
7.1
q + 10 + 8 = 22 . . . 22 dogters/girls sodat/so that q = 4
10
5
10 + 4
1
7.2.1
P(A) = ─── = ──
7.2.2
P(E ∩ X) = ──── = ──
42
21
42
3
10
5
10 + 10
10
7.2.3
P(A ∩ Y) = ─── = ──
7.2.4
P(X ∩ Y) = ───── = ──
42
21
42
21
10
5
4
2
7.2.5
P(A ∩ X') = ─── = ──
7.2.6
P(E ∩ X ∩ Y') = ─── = ──
42
21
42
21
22
11
10
5
7.3.1
P(E) = ─── = ──
7.3.2
P(A ∩ Y) = ─── = ──
42
21
42
21
24
4
24
4
7.3.3
P(X ∪ Y) = ─── = ──
7.3.4
P(X ∩ Y) = ─── = ──
42
7
42
7
10 + 8
9
4
2
7.3.5
P(X') = ──── = ──
7.3.6
P(net/only X) = ─── = ──
42
21
42
21
10
5
7.3.5
P(A ∩ X') = ─── = ──
42
21
7.4
Ja, hulle sny mekaar nie OF hulle het nie gemeenskaplike elemente nie
OF
n(A ∩ E) = 0 OF A ∩ E = ∅
Yes, they do not intersect OF they have no common or shared elements OF
n(A ∩ E) = 0 OF A ∩ E = ∅
8.1
P(A ∩ B) = 0
8.2
P(A ∩ B) = P(A) + P(B)
1
11
6 + 11
= ── + ─── = ──────
A en B onderling uitsluitend en dus geen
4
24
24
gemeenskaplike elemente nie
A and B are mutually exclusive and thus
17
= ───
have no common elements.
24
9.1
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
9.2
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
13
2
7
3
2
7
─── = ─── + ─── − P(A ∩ B)
── = ── + ─── − P(A ∩ B)
15
5
15
5
5
15
6 + 7
13
6 + 7
9
P(A ∩ B) = ───── − ───
P(A ∩ B) = ───── − ───
15
15
15
15
4
= 0
= ───
15
10.1
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
10.2
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
37
2
3
2
3
2
─── = ─── + ─── − P(A ∩ B)
P(A ∪ B) = ── + ─── − ───
56
7
8
9
11
33
16 + 24
37
22 + 27 − 6
P(A ∩ B) = ────── − ───
P(A ∪ B) = ─────────
56
56
99
3
43
= ───
= ───
56
99
10.3
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
1
3
5
── = ─── + P(B) − ───
2
8
24
1
3
5
P(B) = ─── − ─── + ───
2
8
24
12 − 9 + 5
1
= ─────── = ──
24
11.1
A = {2; 3; 5; 7; 11}
B = {2; 4; 6; 8; 10; 12}
C = {1; 2; 3; 4; 6; 12}
5
1
11.2.1
P(A) = ──
11.2.2
P(A ∩ B) = ──
12
12
10
5
8
2
11.2.3
P(A ∪ B) = ── = ──
11.2.4
P(B ∪ C) = ── = ──
12
6
12
3
2
1
1
11.2.5
P(A ∩ C) = ── = ──
11.2.6
P(A ∩ B ∩ C) = ──
12
6
12
1
4
1
11.3.1
P(7) = ──
11.3.2
P(> 8) = ── = ──
12
12
3
3
1
3
1
11.3.3
P(B ∩ (> 7)) = ── = ──
11.3.4
P(faktore/factors 15) = ── = ──
12
4
12
4
3
1
3
1
11.3.5
P(B ∩ (> 7)) = ── = ──
11.3.6
P(faktore/factors 15) = ── = ──
12
4
12
4