Probability Distributions (DP IB Maths: AA HL)

Exam Questions

4 hours31 questions

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	$\frac{1}{15}$	$\frac{2}{15}$	$\frac{a}{15}$	$\frac{b}{15}$

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	$\frac{1}{k}$	$\frac{7}{k^{2}}$	$\frac{4}{k^{2}}$	$\frac{2}{k^{2}}$	$\frac{1}{k^{2}}$	$\frac{1}{k^{2}}$

$x$	$- 5$	$- 3$	$- 1$	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