Events in Probability

Events in Probability – An event is a set of outcomes of a random experiment. It is any subset of the sample space.

Example:
If a die is rolled, the sample space is:

$S = {1, 2, 3, 4, 5, 6}$

Let A be the event “getting an even number”, then:

$A = {2, 4, 6}$

Here, A is an event.

Types of Events in Probability

The different types of events in probability include:

Simple Event and Compound Event

Impossible and Sure Events

Dependent and Independent Events

Mutually Exclusive Events

Equally Likely Events

Complementary event

Exhaustive Events

Simple Event

Any event that comprises a single result from the sample space is known as a simple event. Contains only one outcome.

Example:
Getting a 3 when a die is rolled →

${3}$

Compound (or Composite) Event

Contains more than one outcome of a sample space.

Example:
Getting an odd number →

${1, 3, 5}$

Impossible Event

An event which cannot happen under any circumstances is called an impossible event. Its probability is 0 and can be described by an empty set $\emptyset$ .

Example:
Getting an 8 on a standard 6-sided die.

Meme vibe Example:
Me scoring 100% in maths…
Yeah, impossible (for me at least 🤣)

Sure Event

Occurrence of the event is certain or universal truth. Such an event which has a probability of 1.

Example:
Getting a number less than 7 when rolling a 6-sided die.

Meme vibe Example:
Brain: Study
Me: Let me just check one message… 📱📘🤣 (a sure event in my life!)

Dependent Events

When the occurrence of one event affects the occurrence of another subsequent event or dependent events are those in which the probability of an event changes based on previous outcomes.

Formulas:

AND (Intersection): $P (A \cap B) = P (A) \times P (B | A)$ ➡ Multiply by conditional probability.

OR (Union): $P (A \cup B) = P (A) + P (B) - P (A \cap B)$ (But here, $P (A \cap B)$ depends on conditional probability.)

Example:
Scenario:
There are 10 cookies in a jar — 4 chocolate and 6 plain.
You pick one without looking, eat it, then pick another.
If your first cookie was chocolate, now only 3 chocolate cookies remain.
So the probability of picking another chocolate cookie has changed.
🎯 Why? Because the total number of cookies AND number of chocolate cookies both decreased → it’s dependent.

Humorous Dependent Event Example – “The Mobile Trap”:
Scenario:
Event A: I took a small break and picked up my mobile.
Event B: Mom appeared out of nowhere like a ninja.
🤔 Logic Behind It:
If Event A (me using the phone) doesn’t happen, Event B (mom walking in) probably won’t happen either.
But the moment I touch my phone, the probability of mom catching me spikes to 99.99%.
→ That’s a dependent event!

Independent Events

If the occurrence or non-occurrence of one event does not affect the occurrence or non-occurrence of another event. Or Independent Events are those in which the probability of an event remains the same, regardless of previous outcomes.

Formulas:

AND (Intersection): $P (A \cap B) = P (A) \times P (B)$ ➡ Multiply individual probabilities.

OR (Union): $P (A \cup B) = P (A) + P (B) - P (A \cap B)$ ➡ Use general addition rule. or $P (A \cup B) = P (A) + P (B) - P (A) \times P (B)$

Example 1:
Tossing two different coins →
Heads on one doesn’t affect the other.

Example 2:
Drawing a card from a deck, then tossing a coin
Drawing a card doesn’t impact the coin toss.

Meme Vibe Example:
Me: Studies 10 hours
Exam: Let’s test your luck instead. 📚🎲😂
My marks and my study hours—no correlation found.

Mutually Exclusive Events

Two events A and B are said to be mutually exclusive events if they cannot occur at the same time. In this case, sets A and B are disjoint that is $A \cap B = \emptyset$ .

Formulas:

AND (Intersection): $P (A \cap B) = 0$ ➡ Since both cannot happen at once.

OR (Union): $P (A \cup B) = P (A) + P (B)$ ➡ Because $P (A \cap B) = 0$ , no overlap.

Example 1:
Getting Head and Tail on one coin flip = impossible.

Example 2:
Voting for candidates
Event A: Voting for Candidate A
Event B: Voting for Candidate B
→ You can only choose one candidate per vote.

Equally Likely Events

Events are called equally likely if they have the same probability of occurring.

In simple words:
👉 No event is more likely than the other.

Example 1:
Tossing a fair coin
Outcomes: {Heads, Tails}
Each has probability 1/2 → equally likely.

Example 2:
Rolling a fair 6-sided die
Outcomes: {1, 2, 3, 4, 5, 6}
Each has probability 1/6.

Meme-style Example:
You and your sibling standing in front of the last slice of pizza 🍕
Both have equal chances of grabbing it first —
But Orange cat ends up eating it 😭

Complementary Events

Two events are complementary if one must happen and they cannot both happen at the same time.

In probability:
👉 If A is an event, then the complement of A (written as A′ or Aᶜ) is the event “A does not happen”.

Formula:

$P (A) + P (A') = 1$ $P (A') = 1 - P (A)$

Example 1:
Tossing a coin
A = Getting heads
A′ = Not getting heads = Getting tails
$P (A) + P (A') = \frac{1}{2} + \frac{1}{2} = 1$

Example 2:
Rolling a die
A = Getting an even number = {2, 4, 6}
A′ = Getting an odd number = {1, 3, 5}
$P (A) = \frac{3}{6} = 0.5$ → $P (A') = 1 - 0.5 = 0.5$

Meme-style Example:
Event A: Studying
Event A′: Sleeping 😴
Only one can win the night before exams —
And usually, A′ does 🛏️😂

Exhaustive Events

Exhaustive events are those mutually exclusive events that together cover all possible outcomes of an experiment. A set of events is called exhaustive if at least one of them must occur when an experiment is performed.

In simple terms:
All possible outcomes of the experiment are covered.

Key Point:
If events are exhaustive, then:

$P (A_{1} \cup A_{2} \cup \dots \cup A_{n}) = 1$

Example:
If a die is rolled:
Let
A = getting an even number → ${2, 4, 6}$
B = getting an odd number → ${1, 3, 5}$
Then A and B are exhaustive because:
$A \cup B = {1, 2, 3, 4, 5, 6}$ → covers entire sample space
$P (A \cup B) = 1$

This wraps up the theory part. In the next article, we’ll practice some questions — so stay tuned!