Probability – Basic Level

Probability is the measure of the chance of a particular event happening. It is expressed as a number between 0 and 1.

0 means an impossible event.
1 means a certain event.
Values between 0 and 1 indicate varying chances.

🧠 Basic Formula:

Probability (P) of an event E
= (Number of favorable outcomes) ÷ (Total number of possible outcomes)

P(E) = F / N

🔶 Basic Terminology

🎯 Experiment: Any action or trial that produces outcomes (e.g., tossing a coin).
🎯 Sample Space (S): Set of all possible outcomes.
→ Toss a coin: S = {Head, Tail}
→ Roll a die: S = {1, 2, 3, 4, 5, 6}
🎯 Event: A subset of the sample space (e.g., getting an even number)
🎯 Favorable outcomes: Outcomes that satisfy the required condition.

🔶 Types of Events

👉 Sure Event: An event that is certain to happen. (P = 1)
👉 Impossible Event: An event that cannot happen. (P = 0)
👉 Mutually Exclusive Events: Events that cannot occur at the same time.
👉 Exhaustive Events: A complete set of possible outcomes.

🔷 Properties of Probability

0 ≤ P(E) ≤ 1
P(Sample Space) = 1
P(Not E) = 1 – P(E)
If A and B are mutually exclusive events:
P(A or B) = P(A) + P(B)
If A and B are not mutually exclusive:
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

🔶 Common Examples and Formulas

🎲 1. Tossing a Coin
Sample space = {H, T}
P(Head) = 1/2
P(Tail) = 1/2

🎲 2. Tossing 2 Coins
S = {HH, HT, TH, TT} → total = 4
P(Both heads) = 1/4
P(At least one head) = 3/4

🎲 3. Rolling a Die
S = {1, 2, 3, 4, 5, 6} → total = 6
P(even number) = 3/6 = 1/2
P(number > 4) = 2/6 = 1/3

🎲 4. Drawing a Card (from a standard 52-card deck)
Total cards = 52
♠, ♥, ♦, ♣ — each has 13 cards

P(drawing a heart) = 13/52 = 1/4
P(face card) = 12/52 = 3/13
(3 in each suit: J, Q, K)

🎲 5. Picking a ball from a bag
Bag has 3 red, 5 blue, 2 green balls
Total = 10
P(red) = 3/10
P(not blue) = 5/10 = 1/2

🔷 Important Shortcut Rules

🧠 Rule 1: P(Not A) = 1 – P(A)
🧠 Rule 2: For "at least one" type problems, use:
P(at least one success) = 1 – P(no success)
🧠 Rule 3: If events A and B are independent:
P(A and B) = P(A) × P(B)

🔷 Some Standard Results

1️⃣ P(getting at least one head in 3 tosses)
S = 8 outcomes
Only one outcome has no heads (TTT)
So: P(at least one H) = 1 – 1/8 = 7/8

2️⃣ From 2 dice, find the probability that the sum is 7:
S = 36
Favorable = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)} = 6 outcomes
P = 6/36 = 1/6

🔶 Example Questions

🧩 Q1. What is the probability of getting an even number on a dice?

A. 1/2 B. 1/3 C. 2/3 D. 1/6
✅ Answer: A
Even numbers: 2, 4, 6 → 3/6 = 1/2

🧩 Q2. A card is drawn. What is the probability of drawing a king?

A. 1/13 B. 1/4 C. 1/52 D. 4/13
✅ Answer: A
There are 4 kings → P = 4/52 = 1/13

🧩 Q3. A bag has 5 red and 3 green balls. What’s the probability of not picking a red ball?

A. 5/8 B. 3/8 C. 1/2 D. 1/4
✅ Answer: B
P(not red) = P(green) = 3/8

🧩 Q4. If two coins are tossed, what is the probability of getting at least one tail?

A. 1/4 B. 3/4 C. 1/2 D. 2/3
✅ Answer: B
Only one outcome is HH
P(at least one T) = 1 – 1/4 = 3/4

🧠 Tip for Exams:
If a question says “at least one”, always consider the complement “none” and subtract from 1. It saves time and avoids miscounts.

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