Shatin Pui Ying College

F.5 Mathematics Test on Chapter 15

Time allowed : 90 minutes

Name : Class : ( ) Marks : /

Multiple Choice Questions:(@2 marks)

1. Five persons Tom, Peter, Jason, May and Susan sit randomly around a round table. The probability that May does not sit next to Susan is

A. B.

C. D.

2. The probability that Paul, Kitty and Alice can finish a test in one hour are , and respectively. Find the probability that only Paul cannot finish the test.

A. B.

C. D.

3. A bag contains 30 balls numbered 1 to 30. Two balls are drawn without replacement. Find the probability that both balls are divisible by 3.

A. B.

C. D.

4. In a class, 40% of the students are girls.

Given that 25% of the girls and 37.5% of the boys are choir members. If a student is chosen at random, find the probability that the student is NOT a choir member.

A. 0.3 B. 0.325

C. 0.675 D. 0.7

5. An aeroplane has four engines. The plane can operate with at least one engine. Suppose all the engines work independently and each engine has the probability of 0.1 to be malfunction. Find the probability that the plane can operate.

A. 0.0009 B. 0.0099

C. 0.0999 D. 0.9999

4 6. There are 5 questions in a test. 3 of them are true-or-false questions and 2 of them are multiple choice questions with 4 options. If Billy answers all the questions by wild guessing, find the probability that he gets only one answer wrong.

A. B.

C. D.

7. A parachutist is landing on a circular target of 100 m in diameter. What is the probability that he is within 20 m from the centre of the target?

A. B.

C. D.

Answers :

1. 2. 3. 4.

5. 6. 7.

B. Conventional Questions

1. Three cards are drawn at random from a well-shuffled deck of 52 cards one after the other with replacement. Find the probability that

(a) all the three cards are red,

(b) at least one black card is drawn.

(4 marks)

2. A 3-digit number is formed by arranging 2, 3 and 4 at random. Find the probability that the number is greater than 300.

(3 marks)

3. The probability of getting a ‘3’ for a biased die is 0.2. If this biased die is thrown three times, find the probability of getting

(a) three ‘3’,

(b) no ‘3’,

(6 marks)

4. One letter is chosen from each of the words ‘CERTIFICATE’ and ‘MATHEMATICS’, find the probability that the two letters chosen are

(a) vowels,

(b) the same,

(7 marks)

5. Three light bulbs A, B, and C are connected in series in a circuit. The probabilities that the light bulbs are defective are,andrespectively.

(a) Find the probability that

(i) both light bulbs B and C are defective,

(ii) only light bulb A is defective.

(b) Given that the circuit does not work unless all the light bulbs are not defective, find the probability that

(i) the circuit works,

(ii) the circuit does not work given light bulb A is not defective.

(8 marks)

6. In a test, there are 4 multiple choice questions. Each question consists of 4 options. 1 mark will be awarded for each correct answer, but 1 mark will be deducted for each wrong answer. Suppose Wendy attempts all the questions by wild guessing, find the probability that she will get

(a) 0 mark,

(b) negative marks,

(8 marks)

7. There are two aquariums of fishes. In aquarium A, there are 3 goldfish, 4 clown fish and 1 butterfly fish. In aquarium B, there are 4 goldfish, 3 clown fish and 1 butterfly fish. Suppose a fish is randomly picked from aquarium A and put into aquarium B, then two fish is picked at random one by one from aquarium B and put into a new aquarium C. Find the probability that after the process,

(a) there is a butterfly fish in aquarium A,

(b) there is two butterfly fish in aquarium B,

(d) there is no butterfly fish in aquarium B.

(8 marks)

8. Harry and Jane take turns throwing a dice. The one who first gets a ‘6’ will win the game. Suppose Jane throws the dice first.

(a) Find the probability that Jane will win the game in her

(i) first trial,

(ii) second trial,

(iii) third trial.

(b) Find the probability that Jane wins the game in her nth trial.

(7 marks)

9. A factory producing computer chips has employed a QC checker. When a computer chip is defective, the probability that the QC engineer identifies it (a true defective) is 0.95. When the computer chip is not defective, the probability that the QC engineer identify as defective (false defective) is 0.02. Suppose 0.5% of the computer chips produced is defective.

(a) Find the percentage of the computer chips that would show a

(i) true defective result,

(ii) false defective result.

(b) If a computer chip is identified as defective, what is the probability that it is a true defective? (Give your answer correct to 3 significant figures.)

(c) If 1000 computer chips are produced each day, find the expected number of computer chips that are falsely identified as defective? (Give your answer correct to the nearest integer.)

(7 marks)

END OF PAPER

Marking Scheme of Test 15

A. Multiple Choice Questions

1. A

Number of possible seats for May = 4

Number of favourable seats = 2

∴ P(May does not sit next to Susan)

2. A

P(only Paul cannot finish)

= P(Paul cannot finish and Kitty finishes and Alice finishes)

3. C

P(both balls are divisible by 3)

= P(1st ball divisible by 3) ´ P(2nd ball divisible by 3 | 1st ball divisible by 3)

4. C

P(not a choir member)

= 1 – P(choir member)

= 1 – P(‘girl and choir member’ or ‘boy and choir member’)

= 1 – [P(girl) ´ P(choir member | girl) + P(boy) ´ P(choir member | boy)]

5. D

P(the plane can operate) = 1 – P(all engines fail)

6. D

P(he gets one wrong)

= P(‘only one true-or-false wrong’ or ‘only one multiple choice wrong’)

= P(only one true-or-false wrong) + P(only one multiple choice wrong)

7. C

P(within 20 m from the centre)

B. Conventional Questions :

1. (a) ∵ P(one card is red) =

∴ P(all the three cards are red)

= P(‘one card is red’ and ‘one card is red’ and ‘one card is red’)

= P(one card is red) ´ P(one card is red) ´ P(one card is red)

(b) P(at least one black card is drawn)

2. By the counting principle, the total number of possible outcomes is: 3 ´ 2 = 6

There are 4 favourable outcomes: 324, 342, 423 and 432.

∴ P(greater than 300)

3. (a) P(three ‘3’)

(b) P(no ‘3’)

P(exactly two ‘3’)

= P((3, 3, X) or (3, X, 3) or (X, 3, 3))

= P(3, 3, X) + P(3, X, 3) + P(X, 3, 3)

4. By the counting principle, the total number of possible outcomes in choosing one letter from each of the words is: 11 ´ 11 = 121

(a) There are 5 vowels in ‘CERTIFICATE’ and 4 vowels in ‘MATHEMATICS’, by the counting principle, the total number of favourable outcomes is: 5 ´ 4 = 20

∴ P(vowels)

(b) There are 12 favourable outcomes: CC, EE, TT, TT, II, II, CC, AA, AA, TT, TT and EE.

∴ P(the same)

5. (a) (i) P(both light bulbs B and C are defective)

(ii) P(only light bulb A is defective)

= P(A defective and B not defective and C not defective)

(b) (i) P(the circuit works)

= P(light bulbs A,B and C not defective)

(ii) P(circuit does not work | light bulb A is not defective)

= P(either light bulbs B or C or both are defective)

6. (a) By the counting principle, there are 6 possible outcomes to have 2 correct and 2 wrong answers.

∴ P(0 mark) = P(2 correct and 2 wrong answers)

(b) By the counting principle, there are 4 possible outcomes to have 1 correct and 3 wrong answers.

∴ P(negative marks)

= P(‘3 wrong answers’ or ‘4 wrong answers’)

= P(3 wrong answers) + P(4 wrong answers)

= 1 – [P(negative marks) + P(0 mark)]

7. (a) P(butterfly fish in aquarium A)

= P(butterfly fish is not picked)

(b) P(two butterfly fish in aquarium B)

= P(picked butterfly fish in aquarium A and not picked butterfly fish in aquarium B in two draws)

= P(picked butterfly fish in aquarium A and picked butterfly fish in aquarium B in both draws)

(d) P(no butterfly fish in aquarium B)

= P(‘picked butterfly fish in aquarium A and picked two butterfly fish in aquarium B in two draws’ or ‘not picked butterfly fish in aquarium A and picked butterfly fish in aquarium B in one of the two draws’)

= P(picked butterfly fish in aquarium A and picked two butterfly fish in aquarium B in two draws) + P(not picked butterfly fish in aquarium A and picked butterfly fish in aquarium B in one of the two draws)

8. (a)(i) P(first trial)

(ii) P(second trial)

(iii) P(third trial)

(b) P(wins in her nth trial)

= P(wins in her 1st or 2nd or 3rd or … or nth trial or …)

9. (a) (i) P(true defective)

(ii) P(false defective)

(b) P(true defective | identified defective)

(cor. to 3 sig. fig.)

(cor. to the nearest integer)