Probability Distributions | DP IB Maths: AI HL Exam Questions 2021

1a

1 mark

Three biased coins are tossed.

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

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1b

3 marks

For each coin the probability of getting heads is $\frac{2}{3}$ . A random variable, X, is defined as the number of heads when the three coins are tossed.

Complete the following probability distribution table for X:

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

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1c

2 marks

Hence, by inserting the relevant probabilities, represent the probability distribution for X as a

piecewise function in the form

$P (X = x) = f (x) = \{\begin{cases} x = 0 \\ x = 1 \\ x = 2 \\ x = 3 \\ 0 & otherwise \end{cases}$

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1d

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	k²	0.1	2k	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
$P (X = x)$	$\frac{1}{6}$	$\frac{1}{2} p$	$\frac{1}{8}$	$\frac{3}{2} p$	$\frac{1}{12}$	3p

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

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	1.5p	p	$\frac{3}{25}$	$\frac{1}{10}$	$\frac{3}{50}$	$\frac{1}{100}$

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	-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	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

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

x	-5	-3	-1	0	1	3	5
$P (X = x)$	0.24	2k²	0.04	0.12	3k	0.19	0.11

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

Number of pets, x	0	1	2	3	4
$P (X = x)$	0.44	0.21	0.19	p	0.02

Probability Distributions (DP IB Maths: AI HL)

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