Axiomatic Definition of Probability Lecture 19

 Axiomatic Definition of Probability

 Lecture 19

 Axiomatic Definition of Probability

Let S be a sample space having n sample points, i.e., S = {A1, A2,..., Ai,..., An}. P(Ai), where i = 1, 2,..., n, is the real number we assign to each sample point. Then it satisfied the following axioms:

1.      The probability of any event lies between zero and one.

2. The probability of a sure event is unity, and the probability of an impossible event is zero.

For sure event, i.e., S

For impossible event,

 3. If A and B are mutually exclusive events, then

Theorem 5.1: If Ac is the complement of event A , then show that

Proof: The events Ac and A are exhaustive events.

Note: If the experiment is repeated a finite number of times, say n, then

Example 5.9: The probability of passing a student in the 2^nd semester is 0.55. What is the probability of failing a student in the 2nd semester?

Solution: Let A represent passing a student, and Ac is the probability of failing. P(A) = 0.55

Example 5.10: Three coins are tossed once. Find the probability of at least one head.

Solution: Let A represent at least one head & Ac represent no head.

In a single toss of a coin, the probability of no head is 0.5

That’s

The probability of at least one head (A) in 3 tosses can be obtained as:

Theorem 5.2: If A and B are two events defined on a sample space S, then prove the following

Proof: Using venn diagram

i. 

ii.

The addition law of Probability for non-mutually Exclusive events

Statement: If A and B are two non-mutually exclusive events. Then the probability of event A or event B is given below:

Proof: Using Venn diagram

The event AUB can be written as:

The event B can be written as:

LHS of equation 1 and 2 are equal, so equate the RHS

Addition Law of Probability for Mutually Exclusive events

Statement: If A and B are two mutually exclusive events. Then the probability of event A or event B is given below:

Proof: Let S be a sample space having “n” sample points, and let A and B be two events. A consist m1 and B consist m2 sample point. The AUB consists of m1 + m2 sample points.

Example 5.11: Three fair coins are tossed once. Find the probability of

i. At least one head or exactly one head

ii. Exactly one head or two heads

Solution: The sample space for three coins is given below:

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

n(S) = 8

i. Let A represent at least one head and B represent exactly one head.

A= {HHH, HHT, HTH, THH, HTT, THT, TTH}

n(A) = 7

B = {HTT, THT, TTH}

n(B) = 3

ii.                    Let A represent Exactly one head and B represent two heads

A = {HTT, THT, TTH}

n(A) = 3

B = {HHT, HTH, THH}

n(B) = 3

Example 5.12: Two balance dies are rolled. Find the probability of i) the same outcome or the sum is less than 5. ii) the sum is 6 or 11

Solution: The sample space consists of 36 outcomes, tabulated below:

i. Let A represent the same outcomes and B represent the sum is less than 5.

A = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}

n(A) = 6

B = {(1, 1), (1, 2), (1, 3) (2, 1), (2, 2)}

n(B) = 5

ii. Let A represent the sum of 6 and B represent the sum of 11.

A = {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)}

n(A) = 5

B = {(5, 6), (6, 5)}

n(B) = 2

Example 5.13: Two cards are drawn from a deck of 52 playing cards. Find the probability

i. The card is a diamond card or Ace card.

ii. The card is a heart or diamond card.

Solution: The sample space is given by:

i.                    Let A represent diamond card and B represent Ace card

ii. Let C represent heart card and D represent diamond card.

• Read More: Independent & Dependent Events