Compound events

In case of simple events we consider the probability of the happening or not happening of single events. For example, we might be interested in finding out the probability of drawing a red ball from a bag containing to white and 6 red balls. On the other hand, In case of compound events we consider the join occurrence of two or more events. For example, if a bag contains 10 white and 6 red balls and if two successive draws of 3 balls are made. We shall be finding out the probability of betting 3 white balls in the first draw bad 3 red balls in the second draw-we are thus dealing with a compound event.

Exhaustive events

Events are said to be exhaustive when their totality includes all the possible outcomes of a random experiment. For example, while tossing a dice, the possible outcomes are 1, 2, 3, 4, 5, and 6 and hence possible outcomes are:
       

(1, 1)	(1, 2)	(1, 3)	(1, 4)	(1, 5)	(1, 6)
(2, 1)	(2, 2)	(2, 3)	(2, 4)	(2, 5)	(2, 6)
(3, 1)	(3, 2)	(3, 3)	(3, 4)	(3, 5)	(3, 6)
(4, 1)	(4, 2)	(4, 3)	(4, 4)	(4, 5)	(4, 6)
(5, 1)	(5, 2)	(5, 3)	(5, 4)	(5, 5)	(5, 6)
(6, 1)	(6, 2)	(6, 3)	(6, 4)	(6, 5)	(5, 6)

Complementary events

Let there be two events A and B. A is called the complementary event of B (and vice versa) if A and B are mutually exclusive and exhaustive. For example, when a dice is thrown, occurrence of an even number (2, 4, 6,) and odd number (1, 3, 5,) are complementary events.

Simultaneous occurrence of two events A and B is generally written as Ab.

Theorems of probability

There are two important theorems of probability, namely:

1. The addition theorem: and

2. The multiplication theorem.

Addition theorem

The addition theorem states that if two events A and B are mutually exclusive the probability of A and B. symbolically,
p (A or B) = p (A) + P (B).

Prof of the theorem, if an event A can happen in a1 ways and B in a2 ways. Then the number of ways in which either event can happen is a₁ + a₂. If the total number of possibilities is n, then by definition the probability if either the first or the second event happening is

a₁ + a₂/N = a₁/n + a₂/n
 
But, a₁/n = p (A)

And, a₂/n = p (B)

Hence, p (A or B) = P (A) + P (B).

Illustration: - one card is drawn from a standard pack of 52. What is the probability that it is either a Kina or a queen?

Solution; - there are 4 kings and 4 queens in pack of 52 cards.
   
The probability that the card drawn is a king n= 4/52
 
And the probability that the card brawn is a queen = 4/52

Since the events are mutually exclusive the probability that the card drawn is either a king or a queen

= 4/52 + 4/52 = 8/52 = 2/13

Multiplication theorem

This theorem states that if two events A and B are independent, the prod, ability that they both will occur is equal to the product of their individual probability. Symbolically, it A and B ate independent, then
 
P (A and B) = p (A) A x P (B)

The theorem can be extended to three or more independent events.

Thus,

P (A, B and C) = P (A) x p (B) x P (C)

Illustration:- a man wants to marry a girl having qualities; white complexion- the probability of getting such a girl is one in twenty; handsome dowry-the probability of getting this is one in fifty; westernized manners and etiquettes-the probability thee is one in  hundred. Find out the probability of his getting married to such a girl when the possession of these three attributes is independent.

Solution: - probability of a girl with while complexion

= 1/20 = 0.05

Probability of a girl with handsome dowry

=1/50 = 0.02

Probability of a girl with westwmised manners

= 1/100 = 0.01

Since the events are independent, the probability of simultaneous occurrence of all these qualities

= 1/20 x 1/50 x 1/100 = 0.05 x 0.02 x n0.01 = 0.00001.

It we are given n independent events A₁, A₂, A₃ …...A_N with respective probability of occurrence as p₁, p₂, p₃……..…, pn, then the probability of occurrence of at lead one of the n events A₁, A₂, A₃……………., A_n can be determined as follows:

p (happening of at least one of the events) = 1-p (happening of none of the events). The following example shall illustrate the application of the shove principle.

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