Addition theorem on probabilities

Addition theorem on probabilities:

We shall discuss the addition theorem on probabilities for mutually exclusive events and not mutually exclusive events. Addition theorem on probabilities for mutually exclusive events:

If two events A and B are mutually exclusive, the probability of the occurrence of either A or B is the sum of individual probabilities of A and B. ie P (AÈB) = P(A) + P(B).

This is clearly stated in axioms of probability.

 Addition theorem on probabilities for not-mutually exclusive events:

If two events A and B are not-mutually exclusive, the probability of the event that either A or B or both occur is given as

P(AÈB) = P(A) + P(B) – P(A Ç B)

Proof: 

Let us take a random experiment with a sample space S of N sample points.

From the diagram, using the axiom for the mutually exclusive events, we write

  P(AÈB) = P(A) + P(B) – P(AÇB)

 In the case of three events A,B,C,

P(AÈBÈC) = P(A) + P(B) + P(C) – P( AÇB) – P(AÇC) – P(BÇC ) + P ( A Ç B Ç C)

Compound events:

The joint occurrence of two or more events is called compound events. Thus compound events imply the simultaneous occurrence of two or more simple events.

For example, in tossing of two fair coins simultaneously, the event of getting ‘ atleast one head’ is a compound event as it consists of joint occurrence of two simple events.

Namely,

Event A = one head appears ie A = { HT, TH} and Event B = two heads appears ie B = {HH}

Similarly, if a bag contains 6 white and 6 red balls and we make a draw of 2 balls at random, then the events that ‘ both are white’ or one is white and one is red’ are compound events.

The compound events may be further classified as

(1)   Independent event

(2)   Dependent event

Independent events:

If two or more events occur in such a way that the occurrence of one does not affect the occurrence of another, they are said to be independent events.

For example, if a coin is tossed twice, the results of the second throw would in no way

be affected by the results of the first throw.

Similarly, if a bag contains 5 white and 7 red balls and then two balls are drawn one by one in such a way that the first ball is replaced before the second one is drawn. In this situation, the two events, ‘ the first ball is white’ and ‘ second ball is red’ , will be independent, since the composition of the balls in the bag remains unchanged before a second draw is made.

Dependent events:

If the occurrence of one event influences the occurrence of the other, then the second event is said to be dependent on the first.

In the above example, if we do not replace the first ball drawn, this will change the composition of balls in the bag while making the second draw and therefore the event of ‘drawing a red ball’ in the second will depend on event (first ball is red or white) occurring in first draw.

Similarly, if a person draw a card from a full pack and does not replace it, the result of

the draw made afterwards will be dependent on the first draw.