myUASARevision for UASA
More
My progress
Additional Mathematics questions, chapter 5
Nama / NameKelas / ClassTarikh / Date

Probability Distribution

40 marks. Use Practice to check each answer as you go, or Exam to answer everything first. Show your working for Section B. Use the standard normal table or a calculator for Z.

Section A: objective questions

12 marks
Choose the correct answer. Each question carries 1 mark.
1

Question 1

Which of the following is a discrete random variable?
Show answerHide answer
C
A count of defective bulbs takes the values 0, 1, …, 8, which can be counted. Height, time and mass are measurements that lie in an interval, so they are continuous.
For the teacher: Remembering, low difficulty, standard 5.1.2
Textbook p. 144
2

Question 2

A ball is drawn from a box of red and blue balls, its colour is noted and it is returned. This is done 4 times. X is the number of red balls drawn. Which set shows the values of X?
Show answerHide answer
D
There could be no red ball at all, or 1, 2, 3 or 4 red balls. {1, 2, 3, 4} leaves out 0.
For the teacher: Understanding, low difficulty, standard 5.1.1
Textbook p. 143
3

Question 3

A discrete random variable X has P(X = r) = k(r + 1) for r = 0, 1, 2, 3. Find the value of k.
Show answerHide answer
B
The total probability is 1: k(1 + 2 + 3 + 4) = 10k = 1, so k = 1/10.
For the teacher: Applying, medium difficulty, standard 5.1.3
Textbook p. 145
4

Question 4

Which situation is not a binomial distribution?
Show answerHide answer
A
Without replacement the probability of a red card changes after each draw, so the trials are not independent. The other three have a fixed n, two outcomes and a constant p.
For the teacher: Analysing, medium difficulty, standard 5.2.1
Textbook p. 153
5

Question 5

X ~ B(6, 0.3). Find P(X = 2).
Show answerHide answer
C
P(X = 2) = 6C2(0.3)²(0.7)⁴ = 15 × 0.09 × 0.2401 = 0.3241. Without the 6C2 = 15 you get 0.0216.
For the teacher: Applying, low difficulty, standard 5.2.2
Textbook p. 155
6

Question 6

A binomial distribution X ~ B(n, p) has mean 12 and variance 7.2. Find n and p.
Show answerHide answer
A
np = 12 and npq = 7.2, so q = 7.2/12 = 0.6 and p = 0.4. Then n = 12/0.4 = 30. n = 20, p = 0.6 has the right mean but its variance is 20 × 0.6 × 0.4 = 4.8.
For the teacher: Applying, high difficulty, standard 5.2.4
Textbook p. 162
7

Question 7

X ~ B(5, 0.4). Find P(X ≥ 1).
Show answerHide answer
D
P(X ≥ 1) = 1 − P(X = 0) = 1 − (0.6)⁵ = 1 − 0.0778 = 0.9222. 0.0778 is P(X = 0) itself.
For the teacher: Applying, low difficulty, standard 5.2.2
Textbook p. 155
8

Question 8

X ~ B(150, 0.2). Find the standard deviation of X.
Show answerHide answer
B
Variance = npq = 150 × 0.2 × 0.8 = 24, standard deviation = √24 = 4.90. 24 is the variance, 30 is the mean, and 5.48 is √30.
For the teacher: Applying, low difficulty, standard 5.2.4
Textbook p. 162
9

Question 9

Which statement about the graph of a normal distribution X ~ N(μ, σ²) is true?
Show answerHide answer
C
The normal curve is a bell shape symmetrical about x = μ. The total area under it is 1, and it never touches the axis.
For the teacher: Understanding, low difficulty, standard 5.3.1
Textbook p. 166
10

Question 10

X ~ N(50, 16). Find the z-score for X = 58.
Show answerHide answer
D
σ = √16 = 4. Z = (58 − 50)/4 = 2. Using 16 as σ gives 0.5.
For the teacher: Applying, low difficulty, standard 5.3.3
Textbook p. 172
11

Question 11

Z ~ N(0, 1). Find P(Z > 1.5).
Show answerHide answer
A
The standard normal table gives the right tail, P(Z > 1.5) = 0.0668. The other area, P(Z < 1.5) = 1 − 0.0668 = 0.9332, is the wrong tail.
For the teacher: Applying, low difficulty, standard 5.3.4
Textbook p. 174
12

Question 12

The masses of pineapples are normally distributed with mean 60 and variance 25. Find the probability that a pineapple has a mass of more than 70.
Show answerHide answer
B
σ = 5. Z = (70 − 60)/5 = 2. P(Z > 2) = 0.0228 from the table. 0.9772 is P(Z < 2), the other tail, and 0.1587 comes from using σ = 10.
For the teacher: Applying, medium difficulty, standard 5.3.4
Textbook p. 176

Section B: structured questions

28 marks
Answer all questions. Show your working.
13

Question 13

The probability distribution of a discrete random variable X is shown in the table, where k is a constant.
r0123
P(X = r)0.12kk0.3
[2 marks] (a)Find the value of k.
Show answerHide answer
Tick each point you got.
[1 mark] (b)Find P(X ≥ 2).
Show answerHide answer
Tick each point you got.
For the teacher: Applying, low difficulty, standard 5.1.3
Textbook p. 145
14

Question 14

20% of the pupils in a school walk to school. 8 pupils are chosen at random. X is the number of pupils who walk to school.
[1 mark] (a)State the distribution of X.
Show answerHide answer
Tick each point you got.
[2 marks] (b)Find the probability that exactly 3 pupils walk to school.
Show answerHide answer
Tick each point you got.
[2 marks] (c)Find the probability that at least 2 pupils walk to school.
Show answerHide answer
Tick each point you got.
For the teacher: Applying, low difficulty, standard 5.2.2
Textbook p. 157
15

Question 15

35% of the visitors to a museum are students. 200 visitors are chosen at random.
[1 mark] (a)Estimate the number of students among the 200 visitors.
Show answerHide answer
Tick each point you got.
[2 marks] (b)Find the variance and the standard deviation.
Show answerHide answer
Tick each point you got.
For the teacher: Applying, low difficulty, standard 5.2.4
Textbook p. 162
16

Question 16

15% of the mangoes from a farm are bruised. Find the least number of mangoes that must be checked so that the probability of finding at least one bruised mango is more than 0.9.
[3 marks] (a)Write the inequality for n and solve it.
Show answerHide answer
Dividing by log 0.85, which is negative, reverses the inequality. Round up.
Tick each point you got.
For the teacher: Applying, high difficulty, standard 5.2.5
Textbook p. 163
17

Question 17

The heights of 400 pupils are normally distributed with a mean of 165 cm and a standard deviation of 6 cm.
[2 marks] (a)Find the probability that a pupil is taller than 171 cm.
Show answerHide answer
Tick each point you got.
[2 marks] (b)Find the probability that a pupil's height is between 159 cm and 177 cm.
Show answerHide answer
Tick each point you got.
[1 mark] (c)Estimate the number of pupils shorter than 153 cm.
Show answerHide answer
Tick each point you got.
For the teacher: Applying, low difficulty, standard 5.3.4
Textbook p. 174
18

Question 18

The marks of a test are normally distributed with a mean of 62 and a standard deviation of 10. The top 10% of the candidates receive a distinction.
[1 mark] (a)State the value of z such that P(Z > z) = 0.1.
Show answerHide answer
Tick each point you got.
[3 marks] (b)Find the minimum mark for a distinction.
Show answerHide answer
Tick each point you got.
For the teacher: Applying, medium difficulty, standard 5.3.5
Textbook p. 181
19

Question 19

Aina says: "A bag has 4 red and 6 blue counters. I take out 3 counters one by one without putting them back. The number of red counters, X, has the distribution B(3, 0.4)."
[1 mark] (a)Is Aina correct? Explain.
Show answerHide answer
Tick each point you got.
[1 mark] (b)State how the experiment can be changed so that Aina's distribution is correct.
Show answerHide answer
Tick each point you got.
[1 mark] (c)With the change in (b), find P(X = 1).
Show answerHide answer
Tick each point you got.
For the teacher: Evaluating, medium difficulty, standard 5.2.1
Textbook p. 153
20

Question 20

Create your own situation that has the distribution B(6, 0.3).
[2 marks] (a)Describe the situation and state what X means.
Show answerHide answer
A biased dice is thrown 6 times. The probability of getting a 6 is 0.3. X is the number of 6s.
Tick each point you got.
For the teacher: Creating, medium difficulty, standard 5.2.1
Textbook p. 153