Properties of Probability

Probability is a branch in mathematics that specifies how likely an event can occur. The value of probability is between 0 and 1. Zero(0) indicates an impossible event and One(1) indicates certainly (surely) that will happen. There are a few properties of probability those are mentioned below-

Properties of Probability

1. The probability of an event can be defined as the Number of favorable outcomes of an event divided by the total number of possible outcomes of an event.

Probability(Event)=(Number of favorable outcomes of an event) / (Total Number of possible outcomes)

Example: What is the probability of getting a Tail when a coin is tossed?

Solution:

Number of Favorable Outcomes- {Tail} = 1
Total Number of possible outcomes- {Head, Tail} - 2
Probability of getting Tail= 1/2 = 0.5

2. Probability of a sure/certain event is 1.

Example: What is the probability of getting a number between 1 and 6 when a dice is rolled?

Solution:

Number of favorable outcomes- {1,2,3,4,5,6} = 6
Total Possible outcomes- {1,2,3,4,5,6} = 6
Probability of getting a number between 1 to 6= 6/6 = 1
Probability is 1 indicates it is a certain event.

3. Probability of an impossible event is zero (0).

Example: What is the probability of getting a number greater than 6 when a dice is rolled?

Solution:

Number of favorable outcomes - {} = 0
Total possible outcomes - {1,2,3,4,5,6} = 6
Probability(Number>6) = 0/6 = 0
Probability Zero indicates impossible event.

4. Probability of an event always lies between 0 and 1. It is always a positive. 

0 <= Probability(Event) <= 1

Example: We can notice that in all the above examples probability is always between 0 & 1.

5. If A and B are 2 events that are said to be mutually exclusive events then P(AUB) = P(A) + P(B).

Note: Two events are mutually exclusive when if 2 events cannot occur simultaneously. 

Example: Probability of getting head and tail when a coin is tossed comes under mutual exclusive events.

Solution:

To solve this we need to find probability separately for each possibility. 
i.e, Probability of getting head and Probability of getting tail and sum of those to get P(Head U Tail). 
P(Head U Tail)= P(Head) + P(Tail) = (1/2)+(1/2) = 1

6. Elementary event is an event that has only one outcome. These events are also called atomic events or sample points. The Sum of probabilities of such elementary events of an experiment is always 1.

Example: When we are tossing a coin the possible outcome is head or tail. These individual events i.e. only head or only tail of a sample space are called elementary events.

Solution: 

Probability of getting only head=1/2
Probability of getting only tail=1/2
So, sum=1.

7. Sum of probabilities of complementary events is 1.

P(A)+P(A’)=1

Example: When a coin is tossed, the probability of getting ahead is 1/2, and the complementary event for getting ahead is getting a tail so the Probability of getting a tail is 1/2.

Solution:

If we sum those two then,
P(Head)+P(Head')=(1/2)+(1/2)=1
Head'= Getting Tail

8. If A and B are 2 events that are not mutually exclusive events then 

• P(AUB)=P(A)+P(B)-P(A∩B)
• P(A∩B)=P(A)+P(B)-P(AUB)

Note: 2 events are said to be mutually not exclusive when they have at least one common outcome.

Example: What is the probability of getting an even number or less than 4 when a die is rolled?

Solution: 

Favorable outcomes of getting even number ={2,4,6}
Favorable outcomes of getting number<4 ={1,2,3}
So, there is only 1 common outcome between two events so these two events are not mutually exclusive.

So, we can find P(Even U Number<4)= P(Even) + P(Number<4) - P(Even ∩ Number<4)

P(Even)=3/6=1/2
P(Number<4)=3/6=1/2
P(Even ∩ Number<4)=1/6    (Common element)
P(Even U Number<4)=(1/2) +(1/2)-(1/6)=1-(1/6)=0.83

9. If E₁,E₂,E₃,E₄,E₅,………E_N are mutually exclusive events then Probability(E₁UE₂UE₃UE₄UE₅U……UE_N)=P(E₁)+P(E₂)+P(E₃)+P(E₄)+P(E₅)+…….+P(E_N).

Example: What is the probability of getting 1 or 2 or 3 numbers when a die is rolled.

Solution:

Let A be the event of getting 1 when a die is rolled.
Favorable outcome- {1}
Let B be the event of getting 2 when a die is rolled.
Favorable outcome- {2}
Let C be the event of getting 3 when a die is rolled.
Favorable outcome- {3}
No common favorable outcomes. 

So,  A, B, C are mutually exclusive events.

According to above probability rule- P(A U B U C)= P(A) + P(B) + P(C)

P(A)=1/6
P(B)=1/6
P(C)=1/6
P(A U B U C)=(1/6)+(1/6)+(1/6)=3/6=1/2

These are the top basic properties of probability.