AXIOMATIC DEFINITION OF PROBABILITY

Let us consider a sample space S in connection with a random experiment and let A be an event defined on the sample space S. That is, A ≤ S.

Then a real valued function P defined on S is known as a probability measure and P(A) is defined as the probability of A if P satisfies the following axioms :

(i)  P(A) ≥ 0 for every A ⊆ S (subset)

(ii)  P(S)  =  1

(iii)  For any sequence of mutually exclusive events A₁, A₂, A₃,.................

P(A₁UA₂UA₃.............)  =  P(A₁) + P(A₂) + P(A₃) + .............

Practice Problems

Problem 1 :

A number is selected from the first 25 natural numbers. What is the probability that it would be divisible by 4 or 7 ?

Solution :

Let A be the event that the number selected would be divisible by 4 and B, the event that the selected number would be divisible by 7.

Then AuB denotes the event that the number would be divisible by 4 or 7.

Next we note that

A = {4, 8, 12, 16, 20, 24} and B = {7, 14, 21}

whereas S = {1, 2, 3, ……... 25}.

Here, AnB  =  Null set. 

The two events A and B are mutually exclusive and as such we have

P(AuB)  =  P(A) + P(B) -----(1)

Since

P(A)  =  n(A) / n(S)  =  6/25

and 

P(B)  =  n(B) / n(S)  =  3/25

Thus from (1), we have 

P(AuB)  =  6/25 + 3/25

P(AuB)  =  9/25

So, the probability that the selected number would be divisible by 4 or 7 is 9/25 or 0.36

Problem 2 :

A coin is tossed thrice. What is the probability of getting 2 or more heads ?

Solution :

If a coin is tossed three times, then we have the following sample space.

S  =  {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

2 or more heads imply 2 or 3 heads.

Next we note that

If A and B denote the events of occurrence of 2 and 3 heads respectively, then we find that

A  =  { HHT, HTH, THH } and B  =  { HHH }

P(A)  =  n(A)/n(S)  =  3/8

and 

P(B)  =  n(B)/n(S)  =  1/8

As A and B are mutually exclusive, the probability of getting 2 or more heads is

P(AuB)  =  P(A) + P(B)

P(AuB)  =  3/8 + 1/8

P(AuB)  =  0.50

So, the probability of getting 2 or more heads is 0.50.

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