DP IB Maths: AA SL

Topic Questions

4.4 Probability Distributions

1a
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1 mark

Three biased coins are tossed.

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

1b
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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  2 over 3 .

Complete the following probability distribution table for X:

bold italic x 0 1 2 3
bold italic P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis        
1c
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2 marks

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

P left parenthesis X equals x right parenthesis equals f left parenthesis x right parenthesis equals open curly brackets table row blank row blank end table close

1d
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2 marks

Represent the probability distribution for X as a bar chart.

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2
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2 marks

The random variable X has the probability function

P left parenthesis X equals x right parenthesis equals open curly brackets table row cell fraction numerator x over denominator 3 k end fraction end cell cell space space space space x equals 1 comma 2 comma 3 comma 4 comma 5 end cell row 0 cell space space space space otherwise end cell end table close

Show that  k equals 5.

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3a
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2 marks

The random variable X has the probability function

P left parenthesis X equals x right parenthesis equals open curly brackets table row cell k x end cell cell x equals 1 comma 3 comma 5 comma 7 end cell row cell 0 space 1 over k end cell otherwise end table close

Find the value of k.

3b
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2 marks

Find P left parenthesis X greater than 3 right parenthesis.

3c
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1 mark

State, with a reason, whether or not X is a discrete random variable.

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4a
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2 marks

The random variable X has the probability function

P open parentheses X equals x close parentheses equals open curly brackets table row cell 0.23 space space space space space space space space space x equals negative 1 comma 4 end cell row cell k space space space space space space space space space space space space space space space x equals 0 comma 2 end cell row cell 0.13 space space space space space space space space space x equals 1 comma 3 end cell row cell 0 space space space space space space space space space space space space space space space otherwise end cell end table close

Find the value of k.

4b
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2 marks

Construct a table giving the probability distribution of  X.

4c
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1 mark

Find P left parenthesis 0 less or equal than X less than 3 right parenthesis.

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5
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6 marks

A discrete random variable X has the probability distribution shown in the following table:

bold italic x 0 1 2 3 4
bold italic P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis 5 over 24 1 third 1 fourth 1 over 12 1 over 8

Find:

(i)

P left parenthesis X less than 4 right parenthesis

(ii)

P left parenthesis X greater than 1 right parenthesis

(iii)

P left parenthesis 2 less than X less or equal than 4 right parenthesis

(iv)
P left parenthesis 0 less than X less than 4 right parenthesis

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6a
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1 mark

Leonardo has constructed a biased spinner with six sectors labelled 0,1 , 2, 3, 4 and 5.  The probability of the spinner landing on each of the six sectors is shown in the following table:

number on sector 0 1 1 2 3 5
probability 6 over 20 p 3 over 20 5 over 20 3 over 20 1 over 20

Find the value of p.

6b
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3 marks

Leonardo is playing a game with his biased spinner.  The score for the game,X , is the number which the spinner lands on after being spun.

Leonardo plays the game twice and adds the two scores together. Find the probability that Leonardo has a total score of 5.

6c
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2 marks

Complete the following cumulative probability function table for X:

Score bold space bold italic x 0 1 2 3 5
bold italic P bold left parenthesis bold italic X bold less or equal than bold italic x bold right parenthesis 6 over 20       1

6d
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2 marks

Find the probability that is X

(i)

no more than 1

(ii)
at least 3.

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7a
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2 marks

A discrete random variable X has the following probability distribution:

bold italic x negative 3 negative 1 0 1 3
bold italic P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis 0.11 k squared 0.1 2 k 0.1

where k is a positive constant.

Show that k squared plus 2 k minus 0.69 equals 0.

7b
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1 mark

Hence find the value of k

7c
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3 marks

Find E left parenthesis X right parenthesis.

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8
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4 marks

A spinner is spun on a circle that is divided up into five sections, A comma B comma C comma D and E

The probability of the spinner landing on each section is given by the following table:

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

A person who rotates the spinner scores points depending on which section the spinner lands on.  These points are shown below.

bold Region A B C D E
bold Points negative 5 2 3 10 k

Given that the game is fair, find the value of k

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9a
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3 marks

A discrete random variable X has the following probability distribution:

bold italic x 0 1 2 3 4
bold italic P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis 0.1 0.05 a b 0.1

The value of  E open parentheses X close parenthesesequals 2.3.

Show that a spaceand b must satisfy the following two simultaneous equations:

a plus b equals 0.75

2 a plus 3 b equals 1.85

9b
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2 marks

Hence find the value of a spaceand the value of b .

9c
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2 marks

Find P left parenthesis 1 less or equal than X less than 4 right parenthesis.

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