Module title & Course code
Probability, Sets and Complex Numbers
MAS101
Lecturer
Professor David Jordan
Course description
This unit introduces a number of key mathematical topics essential for the understanding of other
modules at all levels in pure and applied mathematics, statistics and probability. The topics covered
include set theory, logic, permutations and combinations, finite and infinite series, probability and
standard distributions, and the concept of complex numbers and their basic properties.
Further
module
only?
no, but
optional
University course content
A Level course content
Module
Venn diagrams
The drawing and calculation of probabilities using Venn diagrams
S1
Unions, intersections, differences, complement.
Elementary probability.
Set intersection, union, complement ideas will have been met but not the actual
words. Mutual exclusivity, independence. Formulae to be remembered:
S1
no, but
optional
S1
no, but
optional
P( A  B)
,
P( B)
P( A  B)  P( A)  P( B) for independence, E ( X ) , E ( X 2 ) ,
Var ( X )  E ( X 2 )  {E ( X )}2 , E (aX  b)  aE ( X )  b ,
P( A  B)  P( A)  P( B)  P( A  B) , P( A | B) 
Var (aX  b)  a 2Var ( X )
Using the Normal distribution to find z-values, probabilities, and unknown
parameters. Formulae to be remembered: Z 
X 

.
Random variables, sample spaces, events, subsets, frequency probability
S1
no, but
optional
Continuous and discrete sample spaces, cumulative probability functions
(discrete and continuous), probability density functions. Formulae to be
remembered: F ( x) 
P( X  x) for discrete distributions,

S1
no, but
optional
x
Formulae for the numbers of combinations,
Binomial expansion
F ( x)   f ( x)dx for continuous distributions
S2
no, but
optional
The Binomial distribution: X ~ B ( n, p ) , P( X  r )  n C r p r (1  p) nr
S2
no, but
optional
C2
no
C1
no
C2
no
FP2
yes
FP1
yes
Use of Pascal’s triangle and nCr understanding/notation. The use of the Binomial
Expansion. Formula to be remembered:
(1  x) n  1  nx 
Sequences and series. Notation. Convergence and
formal series. Finite and infinite series which can
be summed, including arithmetic and geometric.
Standard series expansions, including binomial,
exponential and trigonometric.
n(n  1) 2 n(n  1)( n  2) 3
x 
x  ... for x  1
2!
3!
Arithmetic progressions: nth term and sum of n terms formulae. Formulae to be
n
remembered: Tn  a  (n  1)d , S n  (2a  (n  1)d )
2
Geometric progressions: nth term, sum of n terms and sum to infinity formulae.
Formulae to be remembered: Tn  ar
n 1
a(1  r )
a
, Sn 
, S 
1 r
1 r
n
Power series: Maclaurin and Taylor expansions. Formula to be remembered:

f ( x)  
n 0
Examples proved by induction
f n (a)
( x  a) n
n!
Understanding and using the Principle of Mathematical Induction to prove results
The complex plane, real and imaginary parts,
modulus and argument, complex conjugate.
Addition, multiplication, division.
Using basic complex numbers: manipulation, conjugation, drawing Argand
diagrams
FP1
yes
Polar coordinates, De Moivre's theorem; the roots
of unity. The complex plane, modulus, Triangle
inequalities, sketching some regions in the
complex plane determined by equations and
inequalities
Polar form of complex numbers including the effects of multiplying and dividing
complex numbers geometrically. Use of De Moivre’s Theorem and Euler’s
Formula; using De Moivre’s theorem to express sin/ cos/ tan( n ) in terms of
FP2/FP3 yes
sin/ cos/ tan  and vice-versa. Complex exponentials and finding n-th roots.
i
Formulae to be remembered: re  r (cos   i sin  ) ;
(cos   i sin  ) n  cos n  i sin n , where    and n  ;
1
1
cos n  ( z n  z  n ) sin n  ( z n  z  n )
2
2i
;
Sketching curves and regions using modulus symbols in the Argand diagram
FP2/FP3 yes