WELCOME TO RICHFIELD
DISTANCE LEARNING
BBA/DBA SEMESTER 3: 2023
WORKSHOP
MODULE: BUS621
LECTURER: NORMAN MAHOHOMA
Topic 5: Discrete Probability Distributions
 A probability distribution is a list of all the possible outcomes of a random
variable and their associated probabilities of occurrence.
 Probability distribution functions can be classified as discrete or continuous.
 Discrete probability distributions assume that the outcomes of a random
variable under study can take on only specific (usually integer) values.
 Examples include the following:
 A maths class can have 1, 2, 3, 4, 5 (or any integer) number of students.
 A company can have 0, 1, 2, 3 (or any integer) employees absent on a day.
Binomial Probability Distribution
A discrete random variable follows the binomial distribution if it satisfies the
following four conditions/characteristics/properties:
 The random variable is observed n number of times (this is equivalent to
drawing a sample of n objects and observing the random variable in each one).
 There are only two, mutually exclusive and collectively exhaustive, outcomes
associated with the random variable on each object in the sample.
 Each outcome has an associated probability – p and q.
 The objects are assumed to be independent of each other, meaning that p
remains constant for each sampled object.
Binomial Distribution Calculations
 This probability can be calculated using the binomial probability distribution
formula:
Binomial Distribution Calculations
EXAMPLE
Knox Insurance has found that 20% (one in five) of all insurance policies are
surrendered (cashed in) before their maturity date. Assume that 10 policies are
randomly selected from the company’s policy database. Using this information,
answer the following questions:
a. What is the probability that four of these 10 insurance policies will have been
surrendered before their maturity date?
b. What is the probability that no more than three of these 10 insurance policies will
have been surrendered before their maturity date?
c. What is the probability that at least two out of the 10 randomly selected policies
will be surrendered before their maturity date?
Binomial Distribution Calculations
Binomial Distribution Calculations
b)
Binomial Distribution Calculations
c)
Topic 6: Continuous Probability Distributions
 A continuous random variable can take on any value (as opposed to only discrete
values) in an interval.
Examples include:
• the length of time to complete a task
• the daily distance travelled by a delivery vehicle
Because there are an infinite number of possible outcomes associated with a
continuous random variable, continuous probability distributions are represented
by curves.
The area under the curve between two 𝒙-limits represents the probability that 𝒙
lies within these limits (or interval).
Normal Probability Distribution
The normal probability distribution has the
following properties:
 The distribution is always described by two
parameters: a mean (µ) and a standard deviation (σ).
 It is symmetrical about a central mean value, µ.
 The total area under the curve will always equal one,
since it represents the total sample space
 The probability associated with a particular interval of
𝒙-values is defined by the area under the normal
distribution curve between the limits of 𝒙-values.
Finding Probabilities for x-limits using the z-distribution
To find the probability that 𝒙 lies between 𝒙𝟏 and 𝒙𝟐 , it is necessary to find the
area under the bell-shaped curve between these x-limits.
This is done by converting the 𝒙-limits into limits that correspond to another
normal distribution called the standard normal distribution (or z-distribution) for
which areas have already been worked out.
To use the standard normal (z) table to find probabilities associated with outcomes
of these numeric random variables, each 𝒙-value must be converted into a z-value,
which is then used to read off probabilities.
Formulae to convert 𝒙-value into z-value:
𝑧 =
𝑥−µ
σ
Standard Normal (z) Probability Distribution
The standard normal distribution, with random variable z, has the following
properties:
Probabilities (areas) based on the standard normal distribution can be read from
standard normal tables (the z-table).
When reading values (areas) off the z-table, it should be noted that:
• the z-limit (to one decimal place) is listed down the left column, and the second
decimal position of z is shown across the top row
• the value read off at the intersection of the z-limit (to two decimal places) is the
area under the standard normal curve (i.e. probability) between 0 < z < k.
The Standard Normal Distribution (z) Table
Normal Distribution Calculations – Example
Normal Distribution Calculations – Example
Normal Distribution Calculations – Example
END OF PRESENTATION
THANK YOU