# Probability — Coins, Dice, Cards and Balls Made Simple

Source: https://skills101.in/learn/aptitude/probability — Skills101 free tutorial (Quantitative Aptitude). Updated 2026-11-09.

**Probability** measures how likely something is, from **0** (impossible) to **1** (certain).

> **P(event) = number of favourable outcomes ÷ total number of outcomes**

Also: **P(not A) = 1 − P(A)**.

## 1. Coins

One coin: H or T → 2 outcomes. **n coins → 2ⁿ outcomes.**

### Example 1
*Three coins are tossed. What is the probability of exactly two heads?*

Total = 2³ = 8. Two heads: HHT, HTH, THH = 3 → **3/8**.

### Example 2
*Three coins are tossed. Probability of at least one head?*

Only TTT has no head. P = 1 − 1/8 = **7/8**.

> **"At least one" = 1 − P(none).** The fastest method, again.

## 2. Dice

One die: 6 outcomes. **Two dice: 36 outcomes.** Draw the grid once and you will never get these wrong.

![A 6 by 6 grid of sums for two dice. The six cells where the sum is 7 lie on one diagonal](https://skills101.in/learn-img/aptitude/two-dice-grid.svg)

| Sum | 2 | 3 | 4 | 5 | 6 | **7** | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Ways | 1 | 2 | 3 | 4 | 5 | **6** | 5 | 4 | 3 | 2 | 1 |

### Example 3
*Two dice are thrown. Probability that the sum is 9?* 4/36 = **1/9**.

### Example 4
*Two dice. Probability of a doublet (same number on both)?* 6/36 = **1/6**.

## 3. Playing cards

A deck has **52 cards**: 4 suits of 13 (♠ spades and ♣ clubs are black; ♥ hearts and ♦ diamonds are red). Each suit has A, 2–10, J, Q, K. **Face cards** = J, Q, K → 12 in total.

### Example 5
*One card is drawn. Probability it is a king or a heart?*

Kings = 4, hearts = 13, but the king of hearts is counted twice → 4 + 13 − 1 = 16 → **16/52 = 4/13**.

> **P(A or B) = P(A) + P(B) − P(A and B).** Subtract the overlap.

## 4. Balls in a bag (use combinations)

### Example 6
*A bag has 5 red and 3 blue balls. Two are drawn at random. Probability both are red?*

Ways to pick 2 red = ⁵C₂ = 10. Ways to pick any 2 = ⁸C₂ = 28 → **10/28 = 5/14**.

### Example 7
*Same bag. Probability of one red and one blue?*

⁵C₁ × ³C₁ = 15 → **15/28**.

## 5. Independent events (AND → multiply)

If one event does not affect the other, **P(A and B) = P(A) × P(B)**.

### Example 8
*A can solve a problem with probability 1/2, B with probability 1/3. If both try, what is the probability it gets solved?*

P(neither solves) = (1 − 1/2)(1 − 1/3) = 1/2 × 2/3 = 1/3. P(solved) = 1 − 1/3 = **2/3**.

## 6. Odds

"Odds in favour 3 : 2" means 3 favourable for every 2 unfavourable → P = 3/(3 + 2) = **3/5**.

## Common mistakes

- Counting (1, 2) and (2, 1) as the same outcome for two dice. They are different cells → 36 outcomes, not 21.
- Double-counting the overlap in "A or B" questions.
- Forgetting that a probability can never be more than 1.

## Quick recap

- P = favourable ÷ total; P(not A) = 1 − P(A).
- n coins → 2ⁿ outcomes; two dice → 36; a sum of 7 is the most likely (6 ways).
- Cards: 52, 4 suits, 12 face cards.
- Balls: use ⁿCᵣ for both favourable and total.
- AND (independent) → multiply; OR → add and subtract the overlap.


Practice quiz and free certificate: https://skills101.in/learn/aptitude/probability