*Question*

A discrete random variable X has the probability distribution given by the following table.

x | 0 | 1 | 2 | 3 |

P( | p | \(\frac{1}{4}\) | \(\frac{1}{6}\) | q |

Given that E (X) = \(\frac{19}{12}\) , determine the value of p and the value of q .

**Answer/Explanation**

**Ans:**

*Question*

A biased four-sided die, A, is rolled. Let X be the score obtained when die A is rolled. The

probability distribution for X is given in the following table.

(a) Find the value of p . [2]

(b) Hence, find the value of E (X ) . [2]

A second biased four-sided die, B, is rolled. Let Y be the score obtained when die B is rolled.

The probability distribution for Y is given in the following table.

(c) (i) State the range of possible values of *r *.

(ii) Hence, find the range of possible values of *q *. [3]

(d) Hence, find the range of possible values for E (*Y *) . [3]

Agnes and Barbara play a game using these dice. Agnes rolls die A once and Barbara rolls die B once. The probability that Agnes’ score is less than Barbara’s score is \(\frac{1}{2}\)

(e) Find the value of E (*Y *) . [6]

**Answer/Explanation**

Ans

## Question

The random variable *T* has the probability density function

\[f(t) = \frac{\pi }{4}\cos \left( {\frac{{\pi t}}{2}} \right),{\text{ }} – 1 \leqslant t \leqslant 1.\]

Find

(a) P(*T* = 0) ;

(b) the interquartile range.

**Answer/Explanation**

## Markscheme

(a) Any consideration of \(\int_0^0 {f(x){\text{d}}x} \) *(M1)*

0 *A1**N2*

(b) **METHOD 1**

Let the upper and lower quartiles be *a* and −*a*

\(\frac{\pi }{4}\int_a^1 {\cos \frac{{\pi t}}{2}{\text{d}}t = 0.25} \) *M1*

\( \Rightarrow \left[ {\frac{\pi }{4} \times \frac{2}{\pi }\sin \frac{{\pi t}}{2}} \right]_a^1 = 0.25\) *A1*

\( \Rightarrow \left[ {\frac{1}{2}\sin \frac{{\pi t}}{2}} \right]_a^1 = 0.25\)

\( \Rightarrow \left[ {\frac{1}{2} – \frac{1}{2}\sin \frac{{\pi a}}{2}} \right] = 0.25\) *A1*

\( \Rightarrow \frac{1}{2}\sin \frac{{\pi a}}{2} = \frac{1}{4}\)

\( \Rightarrow \sin \frac{{\pi a}}{2} = \frac{1}{2}\)

\(\frac{{\pi a}}{2} = \frac{\pi }{6}\)

\(a = \frac{1}{3}\) *A1*

Since the function is symmetrical about *t* = 0 ,

interquartile range is \(\frac{1}{3} – \left( { – \frac{1}{3}} \right) = \frac{2}{3}\) *R1*

**METHOD 2**

\(\frac{\pi }{4}\int_{ – a}^a {\cos \frac{{\pi t}}{2}{\text{d}}t = 0.5 = \frac{\pi }{2}\int_0^a {\cos \frac{{\pi t}}{2}{\text{d}}t} } \) *M1A1*

\( \Rightarrow \left[ {\sin \frac{{a\pi }}{2}} \right] = 0.5\) *A1*

\( \Rightarrow \frac{{a\pi }}{2} = \frac{\pi }{6}\)

\( \Rightarrow a = \frac{1}{3}\) *A1*

The interquartile range is \(\frac{2}{3}\) *R1*

*[7 marks]*

## Examiners report

All but the best candidates struggled with part (a). The vast majority either did not attempt it or let *t* = 1 . There was no indication from any of the scripts that candidates wasted an undue amount of time in trying to solve part (a). Many candidates attempted part (b), but few had a full understanding of the situation and hence were unable to give wholly correct answers.

## Question

The probability distribution of a discrete random variable *X* is defined by

\({\text{P}}(X = x) = cx(5 – x),{\text{ }}x = {\text{1, 2, 3, 4}}\) .

(a) Find the value of *c*.

(b) Find *E*(*X*) .

**Answer/Explanation**

## Markscheme

(a) Using \(\sum {{\text{P}}(X = x) = 1} \) *(M1)*

\(4c + 6c + 6c + 4c = 1\,\,\,\,\,(20c = 1)\) *A1*

\(c = \frac{1}{{20}}\,\,\,\,\,( = 0.05)\) *A1**N1*

* *

(b) Using \({\text{E}}(X) = \sum {x{\text{P}}(X = x)} \) *(M1)*

\( = (1 \times 0.2) + (2 \times 0.3) + (3 \times 0.3) + (4 \times 0.2)\) *(A1)*

\( = 2.5\) *A1**N1*

**Notes:** Only one of the first two marks can be implied.

Award ** M1A1A1** if the x values are averaged only if symmetry is explicitly mentioned.

* *

*[6 marks]*

## Examiners report

This question was generally well done, but a few candidates tried integration for part (b).

## Question

John removes the labels from three cans of tomato soup and two cans of chicken soup in order to enter a competition, and puts the cans away. He then discovers that the cans are identical, so that he cannot distinguish between cans of tomato soup and chicken soup. Some weeks later he decides to have a can of chicken soup for lunch. He opens the cans at random until he opens a can of chicken soup. Let *Y* denote the number of cans he opens.

Find

(a) the possible values of *Y* ,

(b) the probability of each of these values of *Y* ,

(c) the expected value of *Y* .

**Answer/Explanation**

## Markscheme

(a) 1, 2, 3, 4 *A1*

* *

(b) \({\text{P}}(Y = 1) = \frac{2}{5}\) *A1*

\({\text{P}}(Y = 2) = \frac{3}{5} \times \frac{2}{4} = \frac{3}{{10}}\) *A1*

\({\text{P}}(Y = 3) = \frac{3}{5} \times \frac{2}{4} \times \frac{2}{3} = \frac{1}{5}\) *A1*

\({\text{P}}(Y = 4) = \frac{3}{5} \times \frac{2}{4} \times \frac{1}{3} \times \frac{2}{2} = \frac{1}{{10}}\) *A1*

* *

(c) \({\text{E}}(Y) = 1 \times \frac{2}{5} + 2 \times \frac{3}{{10}} + 3 \times \frac{1}{5} + 4 \times \frac{1}{{10}}\) *M1*

\( = 2\) *A1*

*[7 marks]*

## Examiners report

Candidates found this question challenging with only better candidates gaining the correct answers. A number of students assumed incorrectly that the distribution was either Binomial or Geometric.

## Question

The random variable *X* has probability density function *f* where

\[f(x) = \left\{ {\begin{array}{*{20}{c}}

{kx(x + 1)(2 – x),}&{0 \leqslant x \leqslant 2} \\

{0,}&{{\text{otherwise }}{\text{.}}}

\end{array}} \right.\]

Sketch the graph of the function. You are not required to find the coordinates of the maximum.

Find the value of *k* .

**Answer/Explanation**

## Markscheme

* A1*

** **

**Note:** Award ** A1** for intercepts of 0 and 2 and a concave down curve in the given domain .

**Note:** Award ** A0** if the cubic graph is extended outside the domain [0, 2] .

**[1 mark]**

\(\int_0^2 {kx(x + 1)(2 – x){\text{d}}x = 1} \) *(M1)*

**Note:** The correct limits and =1 must be seen but may be seen later.

\(k\int_0^2 {( – {x^3} + {x^2} + 2x){\text{d}}x = 1} \) *A1*

\(k\left[ { – \frac{1}{4}{x^4} + \frac{1}{3}{x^3} + {x^2}} \right]_0^2 = 1\) *M1*

\(k\left( { – 4 + \frac{8}{3} + 4} \right) = 1\) *(A1)*

\(k = \frac{3}{8}\) *A1*

*[5 marks]*

## Examiners report

Most candidates completed this question well. A number extended the graph beyond the given domain.

Most candidates completed this question well. A number extended the graph beyond the given domain.

## Question

A continuous random variable X has the probability density function

\[f(x) = \left\{ {\begin{array}{*{20}{c}}

{k\sin x,}&{0 \leqslant x \leqslant \frac{\pi }{2}} \\

{0,}&{{\text{otherwise}}{\text{.}}}

\end{array}} \right.\]

Find the value of *k*.

Find \({\text{E}}(X)\).

Find the median of *X*.

**Answer/Explanation**

## Markscheme

\(k\int_0^{\frac{\pi }{2}} {\sin x{\text{d}}x = 1} \) *M1*

\(k[ – \cos x]_0^{\frac{\pi }{2}} = 1\)

*k* = 1 *A1*

*[2 marks]*

\({\text{E}}(X) = \int_0^{\frac{\pi }{2}} {x\sin x{\text{d}}x} \) *M1*

integration by parts *M1*

\([ – x\cos x]_0^{\frac{\pi }{2}} + \int_0^{\frac{\pi }{2}} {\cos x{\text{d}}x} \) *A1A1*

= 1 *A1*

*[5 marks]*

\(\int_0^M {\sin x{\text{d}}x} = \frac{1}{2}\) *M1*

\([ – \cos x]_0^M = \frac{1}{2}\) *A1*

\(\cos M = \frac{1}{2}\)

\(M = \frac{\pi }{3}\) *A1*

**Note:** accept \(\arccos \frac{1}{2}\)

* *

*[3 marks]*

## Examiners report

Most candidates scored maximum marks on this question. A few candidates found k = –1.

Most candidates scored maximum marks on this question. A few candidates found k = –1.

Most candidates scored maximum marks on this question. A few candidates found k = –1.

## Question

The discrete random variable *X *has probability distribution:

(a) Find the value of *a*.

(b) Find \({\text{E}}(X)\).

(c) Find \({\text{Var}}(X)\).

**Answer/Explanation**

## Markscheme

(a) \(\frac{1}{6} + \frac{1}{2} + \frac{3}{{10}} + a = 1 \Rightarrow a = \frac{1}{{30}}\) *A1*

* *

(b) \({\text{E}}(X) = \frac{1}{2} + 2 \times \frac{3}{{10}} + 3 \times \frac{1}{{30}}\) *M1*

\(= \frac{6}{5}\)

**A1****Note:** Do not award ** FT **marks if

*a*is outside [0, 1].

*[2 marks]*

* *

(c) \({\text{E}}({X^2}) = \frac{1}{2} + {2^2} \times \frac{3}{{10}} + {3^2} \times \frac{1}{{30}} = 2\) *(A1)*

attempt to apply \({\text{Var}}(X) = {\text{E}}({X^2}) – {\left( {{\text{E}}(X)} \right)^2}\) *M1*

\(\left( { = 2 – \frac{{36}}{{25}}} \right) = \frac{{14}}{{25}}\) *A1*

*[3 marks]*

* *

*Total [6 marks]*

## Examiners report

This was very well answered and many fully correct solutions were seen. A small number of candidates made arithmetic mistakes in part a) and thus lost one or two accuracy marks. A few also seemed unaware of the formula \({\text{Var}}(X) = {\text{E}}({X^2}) – {\text{E}}{(X)^2}\) and resorted to seeking an alternative, sometimes even attempting to apply a clearly incorrect \({\text{Var}}(X) = \sum {{{({x_i} – \mu )}^2}} \).

## Question

Find the coordinates of the point of intersection of the planes defined by the equations \(x + y + z = 3,{\text{ }}x – y + z = 5\) and \(x + y + 2z = 6\).

**Answer/Explanation**

## Markscheme

METHOD 1

for eliminating one variable from two equations *(M1)*

*eg*, \(\left\{ {\begin{array}{*{20}{l}} {(x + y + z = 3)} \\ {2x + 2z = 8} \\ {2x + 3z = 11} \end{array}} \right.\) *A1A1*

for finding correctly one coordinate

*eg*, \(\left\{ {\begin{array}{*{20}{l}} {(x + y + z = 3)} \\ {(2x + 2z = 8)} \\ {z = 3} \end{array}} \right.\) *A1*

for finding correctly the other two coordinates *A1*

\( \Rightarrow \left\{ {\begin{array}{*{20}{l}} {x = 1} \\ {y = – 1} \\ {z = 3} \end{array}} \right.\)

the intersection point has coordinates \((1,{\text{ }} – 1,{\text{ }}3)\)

METHOD 2

for eliminating two variables from two equations or using row reduction *(M1)*

*eg*, \(\left\{ {\begin{array}{*{20}{l}} {(x + y + z = 3)} \\ { – 2 = 2} \\ {z = 3} \end{array}} \right.\) or \(\left( {\begin{array}{*{20}{c}} 1&1&1 \\ 0&{ – 2}&0 \\ 0&0&1 \end{array}\left| {\begin{array}{*{20}{c}} 3 \\ 2 \\ 3 \end{array}} \right.} \right)\) *A1A1*

for finding correctly the other coordinates *A1A1*

\( \Rightarrow \left\{ {\begin{array}{*{20}{l}} {x = 1} \\ {y = – 1} \\ {(z = 3)} \end{array}} \right.\) or \(\left( {\begin{array}{*{20}{c}} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\left| {\begin{array}{*{20}{c}} 1 \\ { – 1} \\ 3 \end{array}} \right.} \right)\)

the intersection point has coordinates \((1,{\text{ }} – 1,{\text{ }}3)\)

METHOD 3

\(\left| {\begin{array}{*{20}{c}} 1&1&1 \\ 1&{ – 1}&1 \\ 1&1&2 \end{array}} \right| = – 2\) *(A1)*

attempt to use Cramer’s rule *M1*

\(x = \frac{{\left| {\begin{array}{*{20}{c}} 3&1&1 \\ 5&{ – 1}&1 \\ 6&1&2 \end{array}} \right|}}{{ – 2}} = \frac{{ – 2}}{{ – 2}} = 1\) *A1*

\(y = \frac{{\left| {\begin{array}{*{20}{c}} 1&3&1 \\ 1&5&1 \\ 1&6&2 \end{array}} \right|}}{{ – 2}} = \frac{2}{{ – 2}} = – 1\) *A1*

\(z = \frac{{\left| {\begin{array}{*{20}{c}} 1&1&3 \\ 1&{ – 1}&5 \\ 1&1&6 \end{array}} \right|}}{{ – 2}} = \frac{{ – 6}}{{ – 2}} = 3\) *A1*

Note: Award *M1 *only if candidate attempts to determine at least one of the variables using this method.

*[5 marks]*

## Examiners report

## Question

The faces of a fair six-sided die are numbered 1, 2, 2, 4, 4, 6. Let \(X\) be the discrete random variable that models the score obtained when this die is rolled.

Complete the probability distribution table for \(X\).

Find the expected value of \(X\).

**Answer/Explanation**

**Answer/Explanation**

## Markscheme

*A1A1*

**Note: **Award ** A1 **for each correct row.

*[2 marks]*

\({\text{E}}(X) = 1 \times \frac{1}{6} + 2 \times \frac{1}{3} + 4 \times \frac{1}{3} + 6 \times \frac{1}{6}\) **( M1)**

\( = \frac{{19}}{6}{\text{ }}\left( { = 3\frac{1}{6}} \right)\) *A1*

**Note: **If the probabilities in (a) are not values between 0 and 1 or lead to \({\text{E}}(X) > 6\) award ** M1A0 **to correct method using the incorrect probabilities; otherwise allow

**marks.**

*FT**[2 marks]*

## Examiners report

[N/A]

[N/A]

## Question

Two unbiased tetrahedral (four-sided) dice with faces labelled 1, 2, 3, 4 are thrown and the scores recorded. Let the random variable *T* be the maximum of these two scores.

The probability distribution of *T* is given in the following table.

Find the value of *a* and the value of *b*.

Find the expected value of *T*.

**Answer/Explanation**

## Markscheme

\(a = \frac{3}{{16}}\) and \(b = \frac{5}{{16}}\) **(M1)A1A1**

**[3 marks]**

**Note:** Award * M1* for consideration of the possible outcomes when rolling the two dice.

\({\text{E}}\left( T \right) = \frac{{1 + 6 + 15 + 28}}{{16}} = \frac{{25}}{8}\left( { = 3.125} \right)\) **(M1)A1**

**Note:** Allow follow through from part (a) even if probabilities do not add up to 1.

**[2 marks]**

## Examiners report

[N/A]

[N/A]

## Question

The continuous variable X has probability density function

\[f(x) = \left\{ {\begin{array}{*{20}{c}}

{12{x^2}(1 – x),}&{0 \leqslant x \leqslant 1} \\

{0,}&{{\text{otherwise}}{\text{.}}}

\end{array}} \right.\]

Determine \({\text{E}}(X)\) .

Determine the mode of *X* .

**Answer/Explanation**

## Markscheme

\({\text{E}}(X) = \int_0^1 {12{x^3}(1 – x){\text{d}}x} \) *M1*

\( = 12\left[ {\frac{{{x^4}}}{4} – \frac{{{x^5}}}{5}} \right]_0^1\) *A1*

\( = \frac{3}{5}\) *A1*

*[3 marks]*

\(f'(x) = 12(2x – 3{x^2})\) *A1*

at the mode \(f'(x) = 12(2x – 3{x^2}) = 0\) *M1*

therefore the mode \( = \frac{2}{3}\) *A1*

*[3 marks]*

## Examiners report

[N/A]

[N/A]

**Question**

**Question**

*[with / without GDC]*

The probability distribution of the discrete random variable X is given by the table

Find

(a) the expected value \(E(X)\) of \(X\).

(b) the mode of \(X\).

(c) the median of \(X\).

(d) the lower quartile \(Q_{1}\) and the upper quartile \(Q_{3}\).

**Answer/Explanation**

**Ans**

(a) \(2.35\) (b) mode =\(1\) (c) median = \(2\) (d) \(Q_{1}\) = \(1\) \(Q_{3}\) = \(3.5\)

**Question**

**Question**

*[without GDC]*

The probability distribution of the discrete random variable X is given by the table

Find the values of \(a\) and \(b\) given that \(E(X)\) = \(2.2\)

**Answer/Explanation**

**Ans**

a = \(0.2\) b = \(0.4\)

**Question**

**Question**

*[without GDC]*

The probability distribution of the discrete random variable X is given by the table.

Nikos selects a number at random.

If he selects \(1\) he earns \(10\) €. If he selects \(2\) he earns \(5\) €. If he selects \(3\) he loses \(4\) €

(a) Find the expected value of \(X\).

(b) Find the expected value of the profit for Nikos. Is the game fair?

**Answer/Explanation**

**Ans**

\(E(X)\) = \(2.2\)\) E(Profit) = \(0\), so it is fair.

**Question**

**Question**

*[without GDC]*

A discrete random variable X has its probability distribution given by

\(P(X=x)=\frac{x}{6}\) where x is \(1, 2, 3\).

(a) Complete the following table showing the probability distribution for \(X\).

(b) Find \(E(X)\).

**Answer/Explanation**

**Ans**

(a)

(b) Find \(E(X)=7/3\)

**Question**

**Question**

*[without GDC]*

Each of the following 10 words is placed on a card and put in a hat.

**ONE, TWO, THREE, FOUR, FIVE, SIX, SEVEN, EIGHT, NINE, TEN**

We pick a card at random. Let \(X\) be the size (number of letters) of the corresponding word.

(a) Give the probability distribution for \(X\) (i.e. the table of probabilities)

(b) Find the expected number of \(X\).

**Answer/Explanation**

**Ans**

(a)

(b) Find \(E(X) = 3.9\)

**Question**

**Question**

*[with GDC]*

The probability distribution of the discrete random variable\(X\) is given by the following table.

(a) Find the value of \(p\).

(b) Calculate the expected value of \(X\).

**Answer/Explanation**

**Ans**

(a) \((0.4 + p + 0.2 + 0.07 + 0.02 = 1), \Rightarrow p = 0.31\)

(b) \(E(X) = 1(0.4) + 2(0.31) + 3(0.2) + 4(0.07) + 5(0.02) = 2\)

**Question**

**Question**

*[without GDC]*

A discrete random variable X has a probability distribution as shown in the table below.

(a) Find the value of \(a + b\).

(b) Given that \(E(X) =1.5\), find the value of \(a\) and of \(b\).

**Answer/Explanation**

**Ans**

(a) \(0.1 + a + 0.3 + b = 1 \Rightarrow a + b = 0.6\)

(b) \(0\times 0.1+1\times a+2\times 0.3+3\times b\)

\(0 + a + 0.6 + 3b = 1.5\)

\(a + 3b = 0.9\)

Solving simultaneously gives

\(a = 0.45\) \(b = 0.15\)

**Question**

**Question**

*[with GDC] *

The following table shows the probability distribution of a discrete random variable \(X\).

(a) Find the value of \(k\).

(b) Find the expected value of \(X\).

**Answer/Explanation**

**Ans**

(a) \(10k^{2}+3k+0.6=1\Rightarrow 10k^{2}+3k+0.4=0\Leftrightarrow k=0.1\) (by GDC)

(b) \(E(X)=-1\times 0.2+2\times 0.4+3\times 0.3=1.5\)

**Question**

**Question**

*[without GDC]*

A discrete random variable \(X\) has its probability distribution given by

\(P(X = x) = k(x + 1)\), where \(x\) is \(0, 1, 2, 3, 4\).

(a) Complete the following table showing the probability distribution for \(X\) (in terms of \(k\))

(b) Show that \(k=\frac{1}{15}\)

(c) Find \(E(X)\).

**Answer/Explanation**

**Ans**

(a)

(b) \(k\times 1+k\times 2+k\times 3+k\times 4+k\times 5=15k=1\Leftrightarrow k=\frac{1}{15}\)

(c) \(E(X)=0\times \frac{1}{15}+1\times \frac{2}{15}+2\times \frac{3}{15}+3\times \frac{4}{15}+4\times \frac{5}{15}=\frac{40}{15}=\frac{8}{3}\)

**Question**

**Question**

*[without GDC]*

** **Three students, Kim, Ching Li and Jonathan each have a pack of cards, from which they select a card at random. Each card has \(a 0, 3, 4\), or \(9\) printed on it.

(a) Kim states that the probability distribution for her pack of cards is as follows.

Explain why Kim is incorrect.

(b) Ching Li correctly states that the probability distribution for her pack of cards is as follows.

Find the value of \(k\).

(c) Jonathan correctly states that the probability distribution for his pack of cards is given by \(P(X=x)=\frac{x+1} {20}\). One card is drawn at random from his pack. Calculate the probability that the number on the card drawn

(i) is \(0\). (ii) is greater than \(0\).

**Answer/Explanation**

**Ans**

(a) Sum = \(1.3\) which is greater than \(1\)

(b) 3k + 0.7 = 1 \Rightarrow k = 0.1

(c) (i) \(P(X=0)=\frac{0+1}{20}=\frac{1}{20}\)

(ii) \(P(X>0)=1-P(X=0)=\frac{19}{20} \left ( or \frac{4}{20}+\frac{5}{20}+\frac{10}{20} \right )=\frac{19}{20}\)

**Notice:** in fact, the probability distribution is

**Question**

**Question**

*[with GDC]*

A fair coin is tossed eight times.

(a) Calculate the probability of obtaining

(b) Find the expected number of heads and the variance of the number of heads

**Answer/Explanation**

**Ans**

(a)

(b)