Business Analytics > Probability, Risk & Sampling > Rules on Probability

Rules on Probability:

1. Probability of an Event

The probability of an event A is the ratio of the number of favorable outcomes to the total number of possible outcomes:

P(A)=Number of favorable outcomes / Total number of outcomes

2. Probability of Complementary Event

The probability that event A does not occur:

P(A^C)=1−P(A)

3. Probability of Union of Events

Union = probability that A or B occurs.

• Mutually exclusive events (cannot happen together): P(A∪B)=P(A)+P(B)
• Non-mutually exclusive events (can happen together): P(A∪B)=P(A)+P(B)−P(A∩B)

4.  Probability of Intersection of Events 

Intersection = probability that A and B both occur:

P(A∩B)=P(A)⋅P(B/A)

If A and B are independent: P(A∩B)=P(A)⋅P(B)

This is often called the multiplication law of probability

Example: Suppose you are going to USA from Bangladesh using a transit in Dubai. From Bangladesh probability of getting the available ticket of “Amirates Air” is 0.7 and from Dubai to USA probability of getting ticket of “Amirates Air” is 0.4. What is the probability that your travel is Dhaka to Dubai via Amirates Air and Dubai to USA also Amirates Air?

Here

A= Amirates Ticket from Dhaka to Dubai = 0.7

B= Amirates Ticket from Dubai to USA = 0.4

Event A and B are independent. So P(Dhaka to Dubai via Amirates Air and Dubai to USA also Amirates Air) is P(A∩B)

P(A∩B) =  P(A)⋅P(B) = 0.7 * 0.4 =0.28

Another Example: Table in the following shows customers gender and and their purchase status. From a dataset this table were generated. Let our interest is on Male who bought  so we can define events as

A= Male and B= Buy

	Buy	Not Buy	Total
Male	6	4	10
Female	14	6	20
Total	20	10	30

P(Male) = 10/30; P(Female) = 20/30; P(Buy) = 20/30; P(Not Buy) = 10/30

i.e. P(A) = 10/30; P(A^c) = 20/30; P(B) = 20/30; P(B^c) = 10/30

P(B/A) = P(Buy within Male) = Male and Buy/Total Male = 6/10

i.e P(B/A) = P(A ∩ B) / P(A)

=> P(A ∩ B) = P(A)⋅P(B/A)

5. Conditional Probability Probability of A given B has occurred:

P(A/B)=P(A∩B) / P(B) given P(B) ) ≠ 0

6. Total Probability Rule

If events B₁, B₂, ..., B_n form a partition of the sample space:

P(A)=P(A∩B₁)+P(A∩B₂)+⋯+P(A∩B_n) or

P(A)= P(A / B₁) . P(B₁) + P(A/  B₂) . P(B₂) + ……. + P(A / B_n). P(B_n)

 

Example: Let our interest is on Male who bought, so we can define events as

A= Male and B= Buy

	Buy	Not Buy	Total
Male	6	4	10
Female	14	6	20
Total	20	10	30

P(Male)=P(Male ∩ Buy) + P(Male ∩ Not Buy)

i.e. P(A) = P(A ∩ B) + P(A ∩ B^C) = 6/30 + 4/30 =10/30
i.e. P(A) = P(A).P(B/A) + P(A). P(A ∩ B^C) = 10/30 . 6/10 + 10/30. 4/10

                                                                    = 6/30 + 4/30 = 10/30

7.    Addition Rule for Three Events

P(A∪B∪C) = P(A) + P(B) + P(C) − P(A∩B) − P(A∩C) − P(B∩C) + P(A∩B∩C)

P(A)= P(A / B₁) . P(B₁) + P(A /  B₂) . P(B₂) + ……. + P(A / B_n). P(B_n)

 

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