Probability: Venn Diagrams & the Addition Rule (Grade 10)

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Grade 10 · Probability

MARKING GUIDELINE

Marks

Duration

55 minutes

Questions

Name:

Class:

Date:

Mark

/ 35

Instructions and Information

Answer ALL the questions in this question paper.
Answer QUESTION 1 by circling the letter (A–D) in the answer grid at the end of that section.
Show ALL calculations clearly.
Show all units where applicable.
Number the answers correctly according to the numbering system used in this question paper.
A non-programmable calculator may be used, unless stated otherwise.
Write neatly and legibly.

Question 1

[10 MARKS]

Four options are given as possible answers to the following questions. Choose the correct answer and circle the letter (A–D) in the grid at the end of this section. If you want to change your choice, put a cross through the wrong letter and circle your new choice.

1.1
A fair die is rolled once. P(an even number) = …
(1)
A)12✓
B)16
C)13
D)23
Answer: A — Three of the six faces — 2, 4 and 6 — are even.
- B — that is the probability of ONE particular face
- C — counted only two even faces instead of three
- D — counted four even faces instead of three
1.2
If P(A) = 0,35 then P(not A) = …
(1)
A)0,65✓
B)0,35
C)1,35
D)0,7
Answer: A — Complementary probabilities add up to 1.
- B — that is P(A) itself
- C — ADDED to 1 instead of subtracting from it
- D — doubled P(A) instead of subtracting it from 1
1.3
Two events are MUTUALLY EXCLUSIVE when …
(1)
A)they cannot happen together, so P(A and B) = 0✓
B)they always happen together
C)P(A) + P(B) = 1
D)P(A) = P(B)
Answer: A — Mutually exclusive events share no outcomes at all.
- B — that is the opposite of mutually exclusive
- C — that describes COMPLEMENTARY events, a special case
- D — equal probabilities say nothing about whether the events overlap
1.4
P(A) = 0,4 and P(B) = 0,3, and A and B are mutually exclusive. Then P(A or B) = …
(1)
A)0,7✓
B)0,12
C)0,1
D)1
Answer: A — For mutually exclusive events P(A or B) = P(A) + P(B), with no overlap to subtract.
- B — MULTIPLIED the probabilities instead of adding them
- C — subtracted instead of adding
- D — assumed the two events fill the whole sample space
1.5
P(A) = 0,5, P(B) = 0,4 and P(A and B) = 0,2. Then P(A or B) = …
(1)
A)0,7✓
B)0,9
C)1,1
D)0,2
Answer: A — P(A or B) = P(A) + P(B) − P(A and B) = 0,5 + 0,4 − 0,2.
- B — forgot to subtract the overlap P(A and B)
- C — ADDED the overlap instead of subtracting it
- D — that is the overlap itself, not the union
1.6
In a class of 60, 8 take History, 41 take Mathematics and 16 take neither. The number taking BOTH is …
(1)
A)5✓
B)11
C)49
D)16
Answer: A — 8 + 41 − both + 16 = 60, so both = 5.
- B — left out the 16 who take neither subject
- C — added 8 and 41 without subtracting the overlap
- D — that is the number taking NEITHER subject
1.7
In that same class, P(a learner takes History ONLY) = …
(1)
A)120✓
B)215
C)112
D)160
Answer: A — History only is 8 − 5 = 3 learners, so the probability is 3/60 = 1/20.
- B — used all 8 History learners instead of only those taking History alone
- C — used the 5 who take both subjects
- D — used one learner instead of three
1.8
In that same class, P(a learner does NOT take Mathematics) = …
(1)
A)1960✓
B)4160
C)1660
D)1941
Answer: A — 60 − 41 = 19 learners do not take Mathematics.
- B — that is P(takes Mathematics), not its complement
- C — that counts only those taking NEITHER, leaving out the History-only learners
- D — divided by 41 instead of by the class total of 60
1.9
Two coins are tossed. The sample space has …
(1)
A)4 outcomes✓
B)2 outcomes
C)3 outcomes
D)8 outcomes
Answer: A — HH, HT, TH and TT — two choices for each of the two coins.
- B — that is the number of outcomes for ONE coin
- C — counted HT and TH as a single outcome
- D — that is the count for THREE coins
1.10
A bag holds 5 red, 3 blue and 2 green marbles. P(not red) = …
(1)
A)12✓
B)310
C)15
D)110
Answer: A — 3 + 2 = 5 marbles are not red, out of 10 altogether.
- B — counted only the blue marbles
- C — counted only the green marbles
- D — used one marble instead of the five that are not red

Answer grid — marking guideline

1.1	A	B	C	D
1.2	A	B	C	D
1.3	A	B	C	D
1.4	A	B	C	D
1.5	A	B	C	D
1.6	A	B	C	D
1.7	A	B	C	D
1.8	A	B	C	D
1.9	A	B	C	D
1.10	A	B	C	D

Question 2

[13 MARKS]

In a school of 70 Grade 10 learners, 32 learners take Economics (E), 43 take Mathematics (M) and 15 take NEITHER of these two subjects. Let the number of learners who take both subjects be x.

2.1
Show that x = 20.
(4)
Learners taking at least one subject: 70 − 15 = 55 (1)
(32 − x) + x + (43 − x) = 55 (2)
75 − x = 55, so x = 20 (1)
2.2
Write down the number of learners who take Economics ONLY and the number who take Mathematics ONLY, and show that all 70 learners are accounted for.
(3)
Economics only = 32 − 20 = 12 (1)
Mathematics only = 43 − 20 = 23 (1)
12 + 20 + 23 + 15 = 70 ✓ — every learner is in exactly one region (1)
2.3
Determine the probability that a learner chosen at random takes BOTH subjects.
(3)
P(E and M) = 2070 (2) = 27 (1)
2.4
Determine the probability that a learner chosen at random takes AT LEAST ONE of the two subjects.
(3)
12 + 20 + 23 = 55 (1)
P = 5570 (1) = 1114 (1)

Question 3

[12 MARKS]

A and B are two events. It is given that P(A) = 0,4 ; P(B) = 0,35 ; P(A and B) = 0,1.

3.1
Calculate P(A or B).
(3)
P(A or B) = P(A) + P(B) − P(A and B) (1)
= 0,4 + 0,35 − 0,1 (1) = 0,65 (1)
3.2
Determine whether A and B are mutually exclusive. Give a reason, and then determine whether they are independent.
(4)
They are NOT mutually exclusive (1)
P(A and B) = 0,1, which is not 0, so the two events CAN happen together (reason 1)
P(A) × P(B) = 0,4 × 0,35 = 0,14 (1)
0,14 ≠ 0,1, so they are not independent either (1)
3.3
Determine whether A and B are independent. Show ALL your working.
(3)
P(A) × P(B) = 0,4 × 0,35 = 0,14 (2)
0,14 ≠ 0,1 = P(A and B), so they are NOT independent (1)
3.4
Calculate the probability that NEITHER A nor B occurs.
(2)
1 − 0,65 (1) = 0,35 (1)

TOTAL: 35 marks

This question paper consists of 3 questions.