In probability theory, mutually exclusive and independent events are fundamental concepts that describe relationships between occurrences. While mutually exclusive events cannot happen simultaneously, independent events do not affect each other’s probabilities. These concepts are crucial for understanding and calculating probabilities in various fields.

In this article, we will discuss in detail mutually exclusive events and independent events and other topics related to them.

What are Mutually Exclusive Events?

Mutually exclusive events are two or more events that cannot occur at the same time or in the same trial. In mathematical terms, this means that the intersection of these events is an empty set (∅). If one event happens, the other(s) cannot happen simultaneously.

In probability theory, we express this as:

P(A and B) = 0

Where,

• A and B are Mutually Exclusive Events
• P(A and B) represents Probability of both A and B Occurring Together

How to Find Mutually Exclusive Events?

To determine if events are mutually exclusive follow the steps added below:

Step 1: Examine the sample space (all possible outcomes).

Step 2: Look at the individual events.

Step 3: Check if there’s any overlap between the events.

Step 4: If there’s no overlap, the events are mutually exclusive.

Examples of Mutually Exclusive Events

Coin toss:

• Event A: Getting Heads
• Event B: Getting Tails

These are mutually exclusive because a coin can’t be both heads and tails in a single toss.

Rolling a Die:

• Event A: Rolling an even number (2, 4, 6)
• Event B: Rolling an odd number (1, 3, 5)

These are mutually exclusive because a number can’t be both even and odd.

Selecting a Card from a Deck:

• Event A: Drawing a Heart
• Event B: Drawing a Spade

These are mutually exclusive because a single card can’t be both a heart and a spade.

Weather:

• Event A: Sunny Day
• Event B: Rainy Day

These are typically considered mutually exclusive (although light rain on a sunny day is possible in some cases).

Rules for Mutually Exclusive Events

• Addition Rule: For mutually exclusive events A and B: P(A or B) = P(A) + P(B). This is different from non-mutually exclusive events, where: P(A or B) = P(A) + P(B) – P(A and B).

• Probability Sum: The sum of probabilities of all mutually exclusive events that make up the entire sample space is always 1 (or 100%).

• Conditional Probability: For mutually exclusive events A and B: P(A|B) = 0 and P(B|A) = 0 This means the probability of A occurring given that B has occurred (and vice versa) is zero.

• Multiplication Rule: For mutually exclusive events A and B: P(A and B) = 0 This is because mutually exclusive events cannot occur together.

• Complementary Events: If A and B are complementary events (mutually exclusive and exhaustive): P(A) + P(B) = 1

What are Independent Events?

Independent events are events where the occurrence of one event does not affect the probability of the other event occurring. In other words, the outcome of one event has no influence on the outcome of the other event.

Mathematically, events A and B are independent if:

P(A|B) = P(A) and P(B|A) = P(B)

where,

• P(A|B) is the Probability of A given that B has occurred
• P(A) is the Probability of A occurring on its Own

How to Find Independent Events?

To determine if events are independent follow the steps added below:

Step 1: Calculate the probability of each event occurring separately: P(A) and P(B).

Step 2: Calculate the probability of both events occurring together: P(A and B).

Step 3: If P(A and B) = P(A) × P(B), then the events are independent.

Examples of Independent events

Coin Tosses:

• Event A: Getting heads on the first toss
• Event B: Getting tails on the second toss

These are independent because the outcome of the first toss doesn’t affect the second.

Rolling Dice:

• Event A: Rolling a 6 on the first die
• Event B: Rolling an even number on the second die

These are independent as the roll of one die doesn’t influence the other.

Drawing Cards with Replacement:

• Event A: Drawing a heart on the first draw
• Event B: Drawing a spade on the second draw (after replacing the first card)

These are independent because replacing the card keeps the probabilities constant.

Weather in Different Cities:

• Event A: Rain in New York
• Event B: Sunshine in Los Angeles

Generally independent, as weather in one city usually doesn’t directly affect another distant city.

Rules for Independent events

Multiplication Rule: For independent events A and B: P(A and B) = P(A) × P(B) This extends to multiple events: P(A and B and C) = P(A) × P(B) × P(C)

Conditional Probability: For independent events A and B: P(A|B) = P(A) and P(B|A) = P(B) The occurrence of one event doesn’t change the probability of the other.

Addition Rule: For independent events A and B: P(A or B) = P(A) + P(B) – P(A) × P(B) This accounts for the overlap in probabilities.

Complement Rule: For independent events A and B: P(not A and not B) = P(not A) × P(not B)

Independence of Complements: If A and B are independent, then:

• A and (not B) are Independent
• (not A) and B are Independent
• (not A) and (not B) are Independent

Pairwise Vs Mutual Independence: Events can be pairwise independent but not mutually independent. For true mutual independence, all possible combinations of events must be independent.

Difference Between Mutually Exclusive Events and Independent Events

Major difference between Mutually Exclusive Events and Independent Events are: