1. Business Statistics Topic 4
Basic Probability and Discrete
Probability Distributions

2. Business Statistics:Topic 4
Learning Objectives
By the end of this topic you will be able to:
Explain basic probability concepts & definitions
Use important rules and procedures for calculating probabilities
Use special discrete probability distributions such as Poisson and Binomial
Business Statistics:Topic 4

3. Business Statistics:Topic 4
Sample Space
The set of all possible outcomes.
e.g: Throwing a dice
The outcomes are {1,2,3,4,5,6}
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4. Business Statistics:Topic 4
Events
Simple Event:
An outcome with only one characteristic from a sample space
e.g A black card from a deck of 52 cards
Joint Event
Two outcomes occurring simultaneously
e.g Picking a black queen from a deck of 52 cards
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5. Business Statistics:Topic 4
Events
Impossible events
e.g drawing a black and red in one card from a deck of 52 cards
Complement of an event A
Denoted by A’
e.g Complement of drawing king of hearts (A)
A’ is all cards in the deck which are not king of hearts
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6. Mutually Exclusive Events
Two events that can not occur simultaneously
e.g: Event A – is drawing king of hearts
Event B – is drawing king of diamonds
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7. Business Statistics:Topic 4
Probability
Probability is the chance of an occurrence of an event (A).
Denoted by p(A) =
e.g Throwing a dice once:
Sample space {1,2,3,4,5,6}
Event A: getting a number less than 3
Possible outcomes: {1,2}
p(A)= 2/6 = 1/3
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8. Business Statistics:Topic 4
Basic Properties
Probability of an event always lies between 0 and 1
The sum of the probabilities of every possible outcome or event is 1.
The probability of the complement A’ is given by 1-P(A)
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9. Probability of Joint events
The probability of joint events A and B is given by P(A  B).
Multiplication rule
e.g. Probability of getting black king
=2/52 =1 /26
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10. Probability Using Contingency Table
P(king black )
= (2/52) = 1/26
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11. Business Statistics:Topic 4
Compound Probability
Probability of getting A or B,
P(A  B) = P(A) + P(B) - P(A  B)
Addition Rule
e.g. Probability of getting a black card or king
=(26 black card + 4 king – 2 black king) / 52
=28/52 = 7/13
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12. Conditional Probability
The probability of an event A given that an event B has occurred, is called conditional probability of A given B.
e.g. probability(black given that it is a king)
=P( getting black and king) / P(king)
=(2/52)/ (4/52) = 2/4 =1/2
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13. Conditional Probability
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14. Probability For Independent Events
Two Events A and B are Independent if:
P(A  B) = P(A)
P(A  B) = P(A) x P(B)
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15. Business Statistics:Topic 4
Random Variable
Numerical outcomes of a random experiment
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16. Discrete Random Variable
Numerical outcomes arising from counting
e.g Throwing a dice three times and counting how many times the number 2 appears.
{0, 1,2 ,3}
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17. Discrete Probability Distribution
The list of possible values of a discrete random variable, Xj and the probabilities, P(Xj)
{ Xj , P(Xj) }
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18. A Discrete Distribution
e.g. tossing two coins and counting the number of times heads appears.
Sample space is {HH,TT,HT,TH}
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19. Business Statistics:Topic 4
Summary Measures
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20. Business Statistics:Topic 4
Expected Mean
Weighted Average of the random variable
e.g Tossing a coin twice and counting the number of heads
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21. Business Statistics:Topic 4
Variance
Weighted average of the squared differences of
the individual values from the mean.
E.g Tossing two coins and counting the number of heads
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22. Business Statistics:Topic 4
Standard Deviation
Positive square root of the variance
In our example
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23. Binomial Probability Distribution
Independent trials are repeated ‘n’ times
e.g tossing a coin more than once
Only two mutually exclusive outcomes on each trial: Success or Failure
e.g tossing a coin: outcomes are Head or
Tail
Constant probability of success for each trial
e.g the probability of getting a head is 1/2 for each toss
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24. Binomial Probability Distribution
Probability of ‘X’ number of successes in ‘n’ independent trails with ‘p’ as the probability of a success
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25. Business Statistics:Topic 4
Example
Tossing a coin three times, n = 3
Success is getting a head
Probability of success is 1/2 = 0.5
Probability of getting 2 heads in 3 tosses, X = 2
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26. Binomial Distribution: Mean and Standard Deviation
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27. Business Statistics:Topic 4
Poisson Distribution
Number of events in a specified interval of time.
Average number of events in the given interval is l
e.g The number of customers arriving in every 10 minutes
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28. Business Statistics:Topic 4
Poisson Distribution
If the average number of events in the given interval is l
The probability of x events occurring in the specified interval is given by:
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29. Business Statistics:Topic 4
Example
Probability of exactly 4 customers arriving in 1 hour when the average per hour is 6.
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30. Poisson Distribution: Mean and Standard Deviation
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