Mathematics questions, chapter 9

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Probability of Combined Events

43 marks. Use Practice to check each answer as you go, or Exam to answer everything first. Show your working for Section B.

Section A: objective questions

15 marks

Choose the correct answer. Each question carries 1 mark.

Question 1

A fair die is rolled and a fair coin is tossed. How many possible outcomes are there?

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n(S) = n(A) × n(B) = 6 × 2 = 12. 8 comes from adding 6 + 2.

For the teacher: Remembering, low difficulty, standard 9.1.1

Textbook p. 245

Question 2

Cards labelled K, A, M are taken out one by one without replacement. How many ordered pairs are in the sample space?

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The first card has 3 choices and the second has 2, so 3 × 2 = 6 pairs. 9 is for choosing with replacement.

For the teacher: Understanding, low difficulty, standard 9.1.1

Textbook p. 245

Question 3

Which pair of events is dependent?

Getting a head on a coin and a 4 on a die
Two red pens taken one by one without replacement from a box of red and blue pens
Getting a tail in two tosses of a coin
Two cards chosen one by one with replacement

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Without replacement the first pen changes what is left in the box, so the second probability changes. The other three are independent.

For the teacher: Understanding, low difficulty, standard 9.2.1

Textbook p. 247

Question 4

A fair coin is tossed and a fair die is rolled. Find the probability of a tail and a number less than 3.

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P(tail) = 12, P(less than 3) = 26 = 13. Product = 16. 112 comes from using only one number.

For the teacher: Applying, low difficulty, standard 9.2.3

Textbook p. 250

Question 5

The probabilities that Ali and Bala pass a test are 0.7 and 0.8 respectively. Find the probability that both pass.

1.5
0.56
0.94
0.15

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Independent events: 0.7 × 0.8 = 0.56. 1.5 is the sum, which cannot be a probability.

For the teacher: Applying, low difficulty, standard 9.2.3

Textbook p. 250

Question 6

A bag contains 4 red and 3 blue balls. Two balls are taken out one by one without replacement. Find P(both red).

1649
47
27
17

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P = 47 × 36 = 1242 = 27. 1649 is the answer with replacement.

For the teacher: Applying, medium difficulty, standard 9.2.3

Textbook p. 251

Question 7

A fair die is rolled. Which pair of events is mutually exclusive?

An even number and a prime number
A number greater than 3 and a number less than 3
An even number and a number greater than 4
An odd number and a prime number

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{4, 5, 6} and {1, 2} share no outcome. The other pairs share 2, 6 and 3 respectively.

For the teacher: Understanding, low difficulty, standard 9.3.1

Textbook p. 253

Question 8

A and B are non-mutually exclusive events. Which is the correct formula for P(A or B)?

P(A) + P(B) − P(A ∩ B)
P(A) × P(B)
P(A) + P(B)
P(A) − P(B)

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Common outcomes are counted twice by P(A) + P(B), so P(A ∩ B) is subtracted. P(A) × P(B) is the rule for independent events 'and'.

For the teacher: Remembering, low difficulty, standard 9.3.2

Textbook p. 256

Question 9

A and B are mutually exclusive events with P(A) = 25 and P(B) = 13. Find P(A or B).

215
38
715
1115

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25 + 13 = 615 + 515 = 1115. 215 is the product, and 38 adds numerators and denominators.

For the teacher: Applying, medium difficulty, standard 9.3.3

Textbook p. 259

Question 10

P(A) = 0.5, P(B) = 0.4 and P(A ∩ B) = 0.15. Find P(A ∪ B).

0.75
0.9
0.2
1.05

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0.5 + 0.4 − 0.15 = 0.75. 0.9 forgets to subtract the intersection.

For the teacher: Applying, low difficulty, standard 9.3.3

Textbook p. 256

Question 11

A box has 3 green and 2 red cards. Two cards are taken without replacement. Find the probability that at least one is red.

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P(none red) = 35 × 24 = 310. P(at least one red) = 1 − 310 = 710. 310 is the probability of no red card.

For the teacher: Applying, medium difficulty, standard 9.3.3

Textbook p. 260

Question 12

A number is chosen from 1 to 12. A is a multiple of 3 and B is an even number. Find P(A or B).

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A = {3, 6, 9, 12}, B = {2, 4, 6, 8, 10, 12}. A ∩ B = {6, 12}. P = 412 + 612 − 212 = 812 = 23. 56 adds without subtracting.

For the teacher: Analysing, high difficulty, standard 9.3.3

Textbook p. 259

Question 13

Two fair dice are rolled. Find the probability that the total is 7 or 11.

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The two totals cannot happen together. Total 7 has 6 outcomes and total 11 has 2: 636 + 236 = 836 = 29.

For the teacher: Applying, medium difficulty, standard 9.3.3

Textbook p. 259

Question 14

Independent events: P(A) = 0.6 and P(B) = 0.25. Find the probability that A happens and B does not happen.

0.15
0.85
0.45
0.4

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P(B') = 1 − 0.25 = 0.75. P(A and B') = 0.6 × 0.75 = 0.45. 0.15 is P(A and B).

For the teacher: Applying, high difficulty, standard 9.3.3

Textbook p. 259

Question 15

Two fair coins are tossed 200 times. Estimate how many times at least one head is obtained.

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P(at least one head) = 1 − 14 = 34. 34 × 200 = 150. 50 is the number of times for two tails.

For the teacher: Evaluating, high difficulty, standard 9.4.1

Textbook p. 262

Section B: structured questions

28 marks

Answer all questions. Show your working.

Question 16

Cards labelled 2, 4 and 7 are in a box. A card is chosen, its number is recorded and it is put back. Then a second card is chosen.

[2 marks] (a)List the sample space.

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All 9 ordered pairs: (2, 2), (2, 4), (2, 7), (4, 2), (4, 4), (4, 7), (7, 2), (7, 4), (7, 7)2

With replacement, the second card has 3 choices every time. Award 1 mark if only 1 or 2 pairs are missing.

Tick each point you got.

[1 mark] (b)State n(S).

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3 × 3 = 9 [1]1

Tick each point you got.

[1 mark] (c)The two events are dependent because the first card is recorded.

ATrueBFalse

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False

The first card is put back, so the box is the same for the second draw. The events are independent.

For the teacher: Understanding, low difficulty, standard 9.1.1

Textbook p. 245

Question 17

A bag has 6 white balls and 4 black balls. Two balls are taken out one by one without replacement. Diagram 1 shows the tree diagram.

Tree diagram. First ball: W 6/10, B 4/10. After W: W 5/9, B 4/9. After B: W is P, B is Q. — Diagram 1

[2 marks] (a)Find the values of P and Q.

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After a black ball is removed, 9 balls remain: 6 white and 3 black [1]1
P = 69 = 23 and Q = 39 = 13 [1]1

Tick each point you got.

[3 marks] (b)Calculate the probability that both balls are the same colour.

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P(W, W) = 610 × 59 = 13 [1]1
P(B, B) = 410 × 39 = 215 [1]1
13 + 215 = 715 [1]1

Tick each point you got.

[2 marks] (c)Calculate the probability that at least one ball is black.

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1 − P(W, W) [1]1
1 − 13 = 23 [1]1

Tick each point you got.

For the teacher: Applying, medium difficulty, standard 9.2.3

Textbook p. 251

Question 18

A card is chosen from cards numbered 1 to 10. A is the event of an even number, B is a multiple of 3 and C is a factor of 10. Diagram 2 shows the Venn diagram.

Venn diagram of 1 to 10. A has 4, 8, 6, 2, 10. C has 2, 10, 1, 5. B has 6, 3, 9. 7 is outside all sets. — Diagram 2

[1 mark] (a)Which two events are mutually exclusive? Give a reason.

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B and C, they have no common element [1]1

Tick each point you got.

[2 marks] (b)Find P(A or B).

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510 + 310 − 110 [1]1
710 [1]1

Tick each point you got.

[2 marks] (c)Find P(A or C).

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A ∩ C = {2, 10}, so 510 + 410 − 210 [1]1
710 [1]1

Tick each point you got.

For the teacher: Analysing, medium difficulty, standard 9.3.2

Textbook p. 257

Question 19

The probability that Aini goes to a camp is 34 and the probability that Bala goes is 23. Their decisions are independent.

[1 mark] (a)Find the probability that both go.

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34 × 23 = 12 [1]1

Tick each point you got.

[2 marks] (b)Find the probability that Aini or Bala goes.

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34 + 23 − 12 [1]1
1112 [1]1

Tick each point you got.

[2 marks] (c)Find the probability that exactly one of them goes.

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1112 − 12 [1] or 34 × 13 + 14 × 231
512 [1]1

Tick each point you got.

For the teacher: Applying, medium difficulty, standard 9.3.3

Textbook p. 259

Question 20

The probability that a bulb made by a factory is defective is 0.1. Two bulbs are chosen at random.

[2 marks] (a)Calculate the probability that at least one bulb is defective.

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P(none defective) = 0.9 × 0.9 = 0.81 [1]1
1 − 0.81 = 0.19 [1]1

Tick each point you got.

[1 mark] (b)500 pairs of bulbs are chosen. Estimate the number of pairs with at least one defective bulb.

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0.19 × 500 = 95 [1]1

Tick each point you got.

[2 marks] (c)A pupil says that the probability of two defective bulbs is 0.2. Explain why the pupil is wrong.

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0.2 is 0.1 + 0.1, which is the addition rule; both must happen, so the events are multiplied [1]1
0.1 × 0.1 = 0.01 [1]1

Tick each point you got.

For the teacher: Evaluating, high difficulty, standard 9.4.1

Textbook p. 262

Question 21

A fair die is rolled. Event A is getting 1, 2 or 3.

[1 mark] (a)Write one event C that is mutually exclusive with A.

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Any set with no element from {1, 2, 3}, for example {5, 6} [1]1

Tick each point you got.

[1 mark] (b)Write one event D that is non-mutually exclusive with A and state the common outcome.

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Any set sharing an element with A, for example D = {3, 4} with common outcome 3 [1]1

Tick each point you got.

For the teacher: Creating, medium difficulty, standard 9.3.1

Textbook p. 253

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