Probability Summary Sheet
Basic Probability
Addition Rule
 The addition rule provides a connection between the probabilities of two events, the
probability of both occurring, and the probability of either occurring.
o 𝑃𝑟 𝐴 ∪ 𝐵 = 𝑃𝑟 𝐴 + 𝑃𝑟 𝐵 − 𝑃𝑟(𝐴 ∩ 𝐵)
Karnaugh Maps
 Karnaugh Maps summarise all the combinations of two events and their complements.
𝑩
𝑷𝒓(𝑨 ∩ 𝑩)
𝑷𝒓(𝑨′ ∩ 𝑩)
𝑷𝒓(𝑩)
𝑨
𝑨′
𝑩′
𝑷𝒓(𝑨 ∩ 𝑩′)
𝑷𝒓(𝑨′ ∩ 𝑩′)
𝑷𝒓(𝑩′)
𝑷𝒓(𝑨)
𝑷𝒓(𝑨′ )
𝟏. 𝟎
Conditional Probability
 The probability of A occurring given that B has occurred is represented by;
o
𝑃𝑟 𝐴 𝐵 =
𝑃𝑟(𝐴∩𝐵)
𝑃𝑟(𝐵)
Independent Events
 If the probability of event A does not influence event B and vice versa, then;
o ∵ 𝑃𝑟(𝐴│𝐵) = Pr(𝐴)
o ∴ 𝑃𝑟 𝐴 ∩ 𝐵 = 𝑃𝑟 𝐴 × 𝑃𝑟 𝐵
Mutually Exclusive Events
 If two events cannot occur simultaneously (mutually exclusive), then;
o ∵ 𝑃𝑟(𝐴 ∩ 𝐵) = 0
o ∴ 𝑃𝑟 𝐴 ∪ 𝐵 = 𝑃𝑟 𝐴 + 𝑃𝑟 𝐵
Combinations
 Combinations refers to the number of ways n different objects can be arranged when
taking r at a time, represented by 𝑛 𝐶𝑟 where;
o
n
Cr =
n!
n−r !r!
Random Variables
Definition of a Random Variable
 A random variable is a function that assigns a numerical value to each outcome of an
experiment.
 A probability distribution consists of all values that a random variable can take, and
their respective probabilities.
 A probability distribution is described by a function p(x) where;
o p x = Pr(X = x)
o 0≤𝑝 𝑥 ≤1
o
𝑝 𝑥 =1
Discrete Random Variables
Discrete Random Variables
 A random variable is said to be discrete if it can only assume a countable number of
values.
 These are typically represented by probability tables such as;
X
a
b
c
d
Pr(X=x)
Pr(X=a)
Pr(X=b)
Pr(X=c)
Pr(X=d)
Expected Value of a Discrete Random Distribution
 The expected value (denoted by E(X)) which is also known as the mean (µ) is given by;
o E X = µ = x ∙ Pr(X = x) where;
 E X + Y = E X + E(Y), where X and Y are discrete random variables
 𝐸[𝑓 𝑥 ] = 𝑓(𝑥) ∙ 𝑝(𝑥)
 E aX + b = aE X + b, where a and b are constants
Median of a Random Distribution
 The median of a random distribution is the value such that 50% of the distribution is
greater than it, and 50% of the distribution is less than it.
o The median (denoted by m) is calculated by adding the probabilities until such
that Pr X ≤ m ≥ 0.5
𝑎+𝑏
o If values exist such that Pr X ≤ a = 0.5, Pr X ≥ b = 0.5, then 𝑚 =
2
Mode of a Random Distribution
 The mode (denoted by M) of a random distribution is the most probable value of the
random variable.
o The mode is such that Pr(𝑋 = 𝑀) ≥ Pr 𝑋 = 𝑥 , for all other values of x
Variance and Standard Deviation
 The variance and standard deviation are measures of spread – that is; how close to the
mean the values of a random probability distribution are.
 The variance of a random variable is defined by;
o σ2 = Var X = E[(X − µ)2 ] = (𝑥 − µ)2 ∙ Pr(𝑋 = 𝑥) where;
 Var aX + b = a2 ∙ Var(X), where a and b are constants
 The standard deviation of a random variable is defined by;
o
𝜎 = 𝑆𝐷 𝑋 =
𝑉𝑎𝑟(𝑋)
Applications of Standard Deviation
 A property of the standard deviation is such that the following ‘confidence intervals’ will
apply
o Pr µ − σ ≤ X ≤ µ + σ ≈ 0.68
o Pr µ − 2σ ≤ X ≤ µ + 2σ ≈ 0.95
o Pr µ − 3σ ≤ X ≤ µ + 3σ ≈ 0.997
Binomial Distributions
Binomial Distributions
 A binomial distribution (also referred to as a Bernoulli sequence) is a particular type of
discrete probability distribution which possesses the following properties:
o Each trial results in only one of two outcomes, one which is denoted as a success,
or one which is denoted as a failure
o The probability of success is constant for all trials
o The trials are independent
Binomial Random Variables
 Binomial random variables are defined by;
o
𝑃𝑟 𝑋 = 𝑥 =
n!
𝑝 𝑥 (1 − 𝑝)𝑛 −𝑥
n−r !r!
where;
 n: Number of trials
 p: Probability of success
 q: Probability of failure
 x: Number of successes
 This can alternatively be expressed as X~Bi (n, p)
Graphs of Binomial Distributions
 The shape of a binomial distribution graph changes according to the values of n and p.
 If p is changed;
o When p<0.5, the graph is positively skewed. i.e. Lower values of x are favoured
o When p=0.5, the graph is symmetrical
o When p>0.5, the graph is negatively skewed. i.e. Higher values of x are favoured
 If n is changed;
o As n increases, the graph becomes increasingly ‘smooth’.
o As n decreases, the graph becomes increasingly ‘rigid’.
Mean for a Binomial Distribution
 In binomial distributions, the mean is calculated by;
 µ=E X =n∙p
 Note: The mode and median are not defined in binomial distributions.
Spread for a Binomial Distribution
 In binomial distributions, the variance is calculated by;
o σ2 = Var X = n ∙ p ∙ 1 − p
Markov Chains
Markov Chains
 Markov Chains possess the following properties;
o The probability of a particular outcome is conditional only on the outcome
before it
o The conditional probabilities of each outcome are the same every time
 Recurrence relationships can be set up to solve Markov chain questions. Examples of
recurrence relationships are;
o ai+1 = Pr 𝑎𝑖+1 𝑎𝑖 ∙ 𝑎𝑖 + Pr 𝑎𝑖+1 𝑏𝑖 ∙ 𝑏𝑖
o bi+1 = Pr 𝑏𝑖+1 𝑏𝑖 ∙ 𝑏𝑖 + Pr 𝑏𝑖+1 𝑎 ∙ 𝑎𝑖
 The long-term behaviour of Markov chains (also known as the steady state) can be
modelled by assuming ai+1 = ai
Continuous Random Variables
Definition of a Continuous Random Variable
 A continuous random variable is one that can take on any value in an interval of the real
number system.
 Due to the infinite number of decimal places to which a continuous random variable can
be measured, a continuous random variable cannot take an exact value as it is rounded
to the limits imposed by the method of measurement used.
o Pr X = x = 0 for all values of x
Continuous Distributions
 Continuous distributions can be represented by a probability density functions (pdf).
Probability density functions are functions that represent continuous distributions, and
as such, possess the following properties;
o f(x) ≥ 0 for all values of x
𝒏
o ∫𝒎 f x dx = 1, where the domain of f(x) is (m,n)
 Both of these properties must be proved to prove a function is probability density
function
 The probability of a continuous random variable falling between x=a and x=b is given
by;
o
b
Pr a ≤ X ≤ b = ∫a f x dx
Mean for a Continuous Distribution
 In a continuous distribution described by the probability density function with domain
[a,b], the mean is calculated by;
o
b
µ = E X = ∫a x ∙ f x dx
b
E[g X ] = ∫a g(x) ∙ f x dx

Percentiles for a Continuous Distribution
 In continuous distributions, the median is calculated by;
o
p
n = ∫a x ∙ f x dx , where p is the nth percentile
To calculate the median, let p=0.5
To calculate the interquartile range (IQR), subtract the 75th percentile
from the 25th percentile
Mode for a Continuous Distribution
 To calculate the mode;


Differentiate the probability density function and find the probability values at
the turning points
o Find the probability values of the end-points of the probability density function.
o Whichever of these has the maximum value of f(x) is the mode
Spread for a Continuous Distribution
 To calculate the variance, the following formula is used in continuous distributions;
o
o
b
b
b
σ2 = Var X = ∫a (x − µ)2 ∙ f x dx = ∫a x 2 ∙ f x dx − ∫a x ∙ f x dx
2
Normal Distributions
The Standard Normal Distribution
 The standard normal distribution is a special type of continuous distribution with a
probability density function which possesses the following properties;
o
It is defined by f x =
1 2
1
e−2 x
2π
o It has a mean, median and mode of 0
o It has a standard deviation of 1
o It is an symmetrical around zero
 The standard normal distribution can be represented as Z~N(0,1)
Normal Distributions
 A normal distribution is similar to the standard normal distribution, except it does not
necessarily have a mean of 0 or standard deviation of 1. Its properties instead are;
o Its median, mode and mean are the same value
o Hence the value of the mean translates the graph horizontally
o It is symmetrical about the mean
o As the value of the standard deviation increases, the graph widens and vice
versa
 Normal distributions are represented by X~N(µ, σ2)
 The graph of a normal distribution can be obtained by the following transformation
upon the graph of the standard normal distribution;
y
o x, y → σx + μ,

σ
This leads to the more general rule;
o
Pr X ≤ x = Pr Z ≤
x−μ
σ