Probability Distributions (DP IB Maths: AA SL)

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	$10$	$k$

Probability Distributions (DP IB Maths: AA SL)

Exam Questions

1. Number & Algebra

1.1 Number Toolkit

1.2 Exponentials & Logs

1.3 Sequences & Series

1.4 Proof & Reasoning

1.5 Binomial Theorem

2. Functions

2.1 Linear Functions & Graphs

2.2 Quadratic Functions & Graphs

2.3 Functions Toolkit

2.4 Further Functions & Graphs

2.5 Transformations of Graphs

3. Geometry & Trigonometry

3.1 Geometry Toolkit

3.2 Geometry of 3D Shapes

3.3 Trigonometry

3.4 Further Trigonometry

3.5 Trigonometric Functions & Graphs

3.6 Trigonometric Equations & Identities

4. Statistics & Probability

4.1 Statistics Toolkit

4.2 Correlation & Regression

4.3 Probability

4.4 Probability Distributions

4.5 Binomial Distributions

4.6 Normal Distributions

5. Calculus

5.1 Differentiation

5.2 Further Differentiation

5.3 Integration

5.4 Further Integration

5.5 Optimisation

5.6 Kinematics

6. Extended Questions

6.1 Extended Questions

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