Probability Distributions (DP IB Maths: AA HL)

Exam Questions

4 hours31 questions

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1 mark

Three biased coins are tossed.

Write down all the possible outcomes when the three coins are tossed.

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3 marks

A random variable, $X$ , is defined as the number of heads when the three coins are tossed.

For each coin the probability of getting heads is $\frac{2}{3}$ .

Complete the following probability distribution table for $X$ :

$x$	0	1	2	3
$P (X = x)$

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2 marks

Represent the probability distribution for $X$ as a piecewise function in the form

$P (X = x) = f (x) = \{\begin{cases} \end{cases}$

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2 marks

Represent the probability distribution for $X$ as a bar chart.

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$x$	0	1	2	3	4
$P (X = x)$	$\frac{5}{24}$	$\frac{1}{3}$	$\frac{1}{4}$	$\frac{1}{12}$	$\frac{1}{8}$

number on sector	0	1	1	2	3	5
probability	$\frac{6}{20}$	$p$	$\frac{3}{20}$	$\frac{5}{20}$	$\frac{3}{20}$	$\frac{1}{20}$

$x$	$- 3$	$- 1$	$0$	$1$	$3$
$P (X = x)$	0.11	$k^{2}$	0.1	$2 k$	$0.1$

$Region$	A	B	C	D	E
$Probability$	0.55	0.15	0.15	$0.1$	$0.05$

$Region$	A	B	C	D	E
$Points$	$- 5$	2	3	$10$	$k$

Score, $x$	1	2	3	4	5	6
$P (X = x)$	$\frac{1}{6}$	$\frac{1}{2} p$	$\frac{1}{8}$	$\frac{3}{2} p$	$\frac{1}{12}$	$3 p$

Region	A	B	C	Missed target
Probability	$\frac{1}{15}$	$\frac{2}{15}$	$\frac{a}{15}$	$\frac{b}{15}$

Region	$A$	$B$	$C$	$D$	$E$	$Miss$
Score $(X)$	1	4	4	8	8	$- 2$
$P (X = x)$	0.3	$p$	$p$	$q$	$q$	0.4

Section	A	B	C	D	E	F
Area	$\frac{2}{7}$	$\frac{5}{21}$	$\frac{4}{21}$	$\frac{1}{7}$	$p$	$\frac{1}{21}$
Prize	$6	$5	$4	$3	$2	$1

Number on sector	1	2	3	4	5	6
probability	$1.5 p$	$p$	$\frac{3}{25}$	$\frac{1}{10}$	$\frac{3}{50}$	$\frac{1}{100}$

Score, $x$	$- 4$	$- 2$	$- 1$	$0$	$1$	$3$	$5$	$6$
$P (X = x)$	$\frac{1}{6}$	$p$	$\frac{1}{8}$	$\frac{1}{4}$	$\frac{1}{12}$	$\frac{1}{8}$	$\frac{1}{48}$	$q$

Prize, $$ s$	0	2	10	20	100	Grand prize
$P (S = s)$	$\frac{1}{2}$	$\frac{1}{4}$	$\frac{1}{6}$	$\frac{1}{24}$	$p$	$\frac{1}{1000}$

$x$	0	1	2	3	4	5	6	7	8
$P (X = x)$	0.32	0.22	0.21	$\frac{1}{k}$	$\frac{7}{k^{2}}$	$\frac{4}{k^{2}}$	$\frac{2}{k^{2}}$	$\frac{1}{k^{2}}$	$\frac{1}{k^{2}}$

Number of apple products, $x$	$P (X = x)$
0	0.1925
1	0.1815
2	0.2250
3	$p$
4	0.0895
5	0.0504
6	0.0307
7	0.0104
8	$q$

$x$	$- 5$	$- 3$	$- 1$	0	1	3	5
$P (X = x)$	0.24	$2 k^{2}$	0.04	0.12	$3 k$	$0.19$	$0.11$

$y$
$P (Y = y)$

$x$	$- 5$	$- 3$	$- 1$	$0$	$1$	$3$	5
$P (X = x)$	0.42	$\frac{k^{4}}{2}$	0.05	0.21	${(\frac{3 k}{4})}^{2}$	$0.09$	$0.07$

$x$	0	1	2	3	4
$P (X = x)$	0.04	0.35	$a$	$0.21$	b

Probability Distributions (DP IB Maths: AA HL)

Exam Questions

1. Number & Algebra

1.1 Number & Algebra Toolkit

1.2 Exponentials & Logs

1.3 Sequences & Series

1.4 Simple Proof & Reasoning

1.5 Further Proof & Reasoning

1.6 Binomial Theorem

1.7 Permutations & Combinations

1.8 Complex Numbers

1.9 Further Complex Numbers

1.10 Systems of Linear Equations

2. Functions

2.1 Linear Functions & Graphs

2.2 Quadratic Functions & Graphs

2.3 Functions Toolkit

2.4 Other Functions & Graphs

2.5 Reciprocal & Rational Functions

2.6 Transformations of Graphs

2.7 Polynomial Functions

2.8 Inequalities

2.9 Further Functions & Graphs

3. Geometry & Trigonometry

3.1 Geometry Toolkit

3.2 Geometry of 3D Shapes

3.3 Trigonometry Toolkit

3.4 Trigonometry

3.5 Trigonometric Functions & Graphs

3.6 Trigonometric Equations & Identities

3.7 Inverse & Reciprocal Trig Functions

3.8 Further Trigonometry

3.9 Vector Properties

3.10 Vector Equations of Lines

3.11 Vector Planes

4. Statistics & Probability

4.1 Statistics Toolkit

4.2 Correlation & Regression

4.3 Probability

4.4 Probability Distributions

4.5 Binomial Distribution

4.6 Normal Distribution

4.7 Further Probability Distributions

5. Calculus

5.1 Differentiation

5.2 Further Differentiation

5.3 Integration

5.4 Further Integration

5.5 Optimisation

5.6 Kinematics

5.7 Basic Limits & Continuity

5.8 Advanced Differentiation

5.9 Advanced Integration

5.10 Differential Equations

5.11 MacLaurin Series

5.12 Further Limits (inc l'Hôpital's Rule)

6. Extended Questions

6.1 Extended Questions (Section B, SL)

6.2 Extended Questions (Section B, HL)