Page 266 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Unit 12: Probability and Expected Value

Theorem 3: Notes
For any two events A and B in a sample space S

P A  B P B P A B
Theorem 4: (Addition of Probabilities)



P A B P A P B P A B
Example: What is the probability of drawing a black card or a king from a well-shuffled
pack of playing cards?
Solution:
There are 52 cards in a pack, n(S) = 52.
Let A be the event that the drawn card is black and B be the event that it is a king. We have to find.

P A B
Since there are 26 black cards, 4 kings and two black kings in a pack, we have n(A) = 26, n(B) = 4
26 4 2 7
and  B = 2 Thus, P A B
52 13
Alternative Method
The given information can be written in the form of the following table:
B B Total
A 2 24 26
A 2 24 26
Total 4 48 52
From the above, we can write

24 7

P A B 1 P A  B 1
52 13
Theorem 5: (Multiplication or Compound Probability Theorem)
A compound event is the result of the simultaneous occurrence of two or more events.
For convenience, we assume that there are two events, however, the results can be easily
generalised. The probability of the compound event would depend upon whether the events are
independent or not. Thus, we shall discuss two theorems: (a) Conditional Probability Theorem,
and (b) Multiplicative Theorem for Independent Events.

1. Conditional Probability Theorem: For any two events A and B in a sample space S, the
probability of their simultaneous occurrence, is given by

P A  B P A P B / A
or equivalently P B P A / B
Here, P(B/A) is the conditional probability of B given that A has already occurred. Similar
interpretation can be given to the term P(A/B).

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