Independent Events in Probability: Definition and Solved Examples

Independent events are one of the many events (Impossible events, Sure Event, Simple Events, Compound Events, Independent Events, Dependent Events, Exhaustive Events, and Complementary Events) in Probability.

Keyterms: Probability, Impossible events, Sure Event, Simple Events, Compound Events, Independent Events, Dependent Events, Exhaustive Events, Complementary Events

Read More: Determinant of a Matrix

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What are Independent Events?

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Independent Events are those events that are not dependent on the happening of any other event. For example, If we flip a dice and get the outcome 2 and if we flip it again and get the outcome 6. In Both the cases, the events have different outcomes and are not dependent on each other. 

All the events that are not dependent on the occurrence and nonoccurrence are termed as independent events. If Event 1 is not dependent on the occurrence of Event 2, then both Event 1 and Event 2 are independent Events. 

Two events 1 and 2 are independent if, 

P(2|1) = P (2) provided P (1) ≠ 0 

and 

P (1|2) = P (1) provided P (2) ≠ 0 

Two events 1 and 2 are also independent if, 

P(1 ∩ 2) = P(1) . P (2)

If we talk about three events such as X,Y and Z are mutually independent if 

P(X ∩ Y) = P (X) P (Y) 

P(X ∩ Z) = P (X) P (Z) 

P(Y ∩ Z) = P (Y) P(Z) 

and 

P(X ∩ Y ∩ Z) = P (X) P (Y) P (Z)

The video below explains this:

Independent Events Detailed Video Explanation:

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Solved Examples

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Example 1: Prove that if X and Y are independent events, then X and Y’ are also independent events.

Ans. Since X and Y are independent, 

we have P(X ∩ Y) = P(X). P(Y) ....(1) 

X ∩ Y and X ∩ Y′ are mutually exclusive events and also X =(X ∩ Y) ∪ (X ∩ Y′). 

Therefore P(X) = P(X ∩ Y) + P(X ∩ Y′) 

or P(X ∩ Y′) = P(X) − P(X ∩ Y) 

= P(X) − P(X) . P(Y) …..by 1

= P(X) (1−P(Y) 

= P(X). P(Y′) 

Hence, X and Y′ are independent

Example 2: If X and Y are two independent events, then prove that the probability of happening of at least one event X and Y is given by 1 - P(X’)P(Y’).

Ans. P(at least one of X and Y) = P(X ∪ Y) 

= P(X) + P(Y) − P(X ∩ Y) 

= P(X) + P(Y) − P(X) P(Y) 

= P(X) + P(Y) [1−P(X)] 

= P(X) + P(Y). P(X′) 

= 1− P(X′) + P(Y) P(X′) 

= 1− P(X′) [1− P(Y)] 

= 1− P(X′) P (Y′)

Example 3: Let A and B be two independent events, where P(A)= 0.1 and P(B)=0.8. Find P(A and B), P(A or B), P( B not A), and P(neither A nor B)

Ans. P(A) = 0.1 and P(B) = 0.8 and events A and B are independent of each other.

P(A and B) = P( A ∩ B) = P(P) P(B) = 0.1 × 0.8 = 0.08

P(A or B) = P(A ∪ B) = P(A) + P(B) – P(A ∩ B) = 0.1 + 0.8 – 0.08 = 0.82

P(B not A) = P(B ∩ A’) = P(B) – P(A ∩ B) = 0.8 – 0.82 = -0.02

And P(neither A nor B) = P(A’ ∩ B’) = 1 – P(A ∪ B) = 1 – 0.82 = 0.18

Example 4 : When given a deck of 52 cards, what is the probability of choosing an ace, spade and a four?

Ans. Probability of getting an Ace = 4/52 = 1/13 since there can be 52 outcomes and a deck of 52 cards consists of 4 aces.

Probability of getting a spade = 13/52 = 1/4 since there can be 52 outcomes and a deck of 52 cards consists of 13 spades.

Probability of getting a four = 4/52 = 1/13 since there can be 52 outcomes and a deck of 52 cards consists of 4 fours.

Example 5: A jar of balls filled with 6 blue balls, 8 red balls, 2 green, and 6 black balls. A ball is chosen at random from the jar and replaced by another ball. Find the probability for P(green and red) and P(blue and black)

Ans. Probability for Getting a green ball= 2/22 = 1/11 since there are 22 balls and 2 green balls.

Probability for Getting a red ball= 8/22 = 4/11 since there are 22 balls and 8 red balls.

Probability for getting a green and red ball = 1/11 * 4/11 = 4/121

Probability for Getting a blue ball= 6/22 = 3/11 since there are 22 balls and 6 blue balls.

Probability for Getting a black ball= 6/22 = 3/11 since there are 22 balls and 6 black balls.

Probability for getting a blue and black balls = 3/11 * 3/11 = 9/121

Independent Events and Mutually Exclusive Events

When talking about Independent events, many people confuse them with mutually exclusive events. 

Mutually Exclusive Events: Events which cannot occur at the same time or simultaneously. Mutually Exclusive Events are also called Disjoint events. For Example, if X and Y are mutually exclusive events, then the probability of these events occurring at the same time is zero. 
If we compare, Independent events and Mutually exclusive events, we can notice that they both have different meanings such as:

1. Independent events are in no way dependent on each other and can take place at the same time but mutually exclusive events cannot occur at the same time.
2. Mutually Exclusive events mean that if one event is not occurring then some other event is occurring meanwhile independent events can take place without affecting one another.

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