# 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 getP(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 findP(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 E_{1},E_{2},E_{3},E_{4},E_{5},………E_{N }are mutually exclusive events then Probability(E_{1}UE_{2}UE_{3}UE_{4}UE_{5}U……UE_{N})=P(E_{1})+P(E_{2})+P(E_{3})+P(E_{4})+P(E_{5})+…….+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)=1/6P(A U B U C)=(1/6)+(1/6)+(1/6)=3/6=1/2

These are the top basic properties of probability.