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CBSE • Class X • Mathematics • Ch 14
Estimated Time: 45 Mins
Study Progress: In Progress

Probability

In Class 10 Mathematics, "Probability" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

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🎲 Have You Ever Wondered?

How do casinos and insurance companies guarantee multibillion-dollar profits without ever cheating? Classical probability calculates exact theoretical...

How do casinos and insurance companies guarantee multibillion-dollar profits without ever cheating? Classical probability calculates exact theoretical chances of random events across coins, dice, and playing cards.

Why This Chapter Matters

In Class 10 Mathematics, "Probability" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

Before You Begin (Prerequisites)

  • Fractions and percentages.
  • Basic chance from Class 7 & 9.
  • Sample spaces.

What You Will Learn (Core Objectives)

  • Define Classical (Theoretical) Probability: $P(E) = \frac{n(E)}{n(S)}$.
  • Identify impossible events ($P = 0$) and sure events ($P = 1$), verifying $0 \le P(E) \le 1$.
  • Apply the Complementary Event rule: $P(E) + P(\text{not } E) = 1$.
  • Analyze sample spaces of flipping 1, 2, and 3 coins, and rolling 1 and 2 dice.
  • Calculate probabilities across a standard 52-card deck (4 suits, 12 face cards).

Chapter Roadmap & Progression

1 1. Theoretical Probability Formula
2 2. The Standard 52-Card Deck Breakd...
3 3. Rolling Two Dice ($n = 36$)

Complete Concept Guide (100% Curriculum Coverage)

1. Theoretical Probability Formula

For equally likely outcomes, the probability of event $E$ is: $$\mathbf{P(E) = \frac{\text{Number of outcomes favorable to } E}{\text{Total number of possible outcomes in sample space } S}}$$
• Range: $0 \le P(E) \le 1$.
• Complementary Rule: $\mathbf{P(E) + P(\bar{E}) = 1}$ (Probability of happening + not happening is strictly $1$).

2. The Standard 52-Card Deck Breakdown

Total: 52 cards split into 4 suits of 13 cards each:
• Red (26): Hearts (13), Diamonds (13).
• Black (26): Spades (13), Clubs (13).
• Face Cards (12): 4 Kings, 4 Queens, 4 Jacks.
• Aces (4): 1 in each suit (not considered face cards!).

3. Rolling Two Dice ($n = 36$)

When rolling two standard dice, there are $6 \times 6 = 36$ total outcomes: $(1, 1), (1, 2)\dots(6, 6)$. Sums range from $2$ (only $(1, 1)$) to $12$ (only $(6, 6)$).

Visual Learning & Conceptual Map

Probability Master Matrix

Conceptual framework, core mechanisms, and analytical relationships
Academic Architecture

1. Theoretical Probability Formula • 2. The Standard 52-Card Deck Breakdown

Chapter Summary & 10 Key Takeaways

Takeaway 1
Probability Formula: $P(E) = \frac{\text{Favorable outcomes}}{\text{Total possible outcomes}}$.
Takeaway 2
Probability Boundaries: Strictly $0 \le P(E) \le 1$.
Takeaway 3
Complementary Rule: $P(E) + P(\text{not } E) = 1$.
Takeaway 4
52-Card Pack: 26 Red, 26 Black, 12 Face cards, 4 Aces.
Takeaway 5
Two Dice: $36$ total sample outcomes.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

1
One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting: (a) A king of red color (b) A face card.
Reveal Answer & Explanation
Answer: (a) There are 2 red kings (hearts and diamonds) $\implies P = \frac{2}{52} = \frac{1}{26}$. (b) There are 12 face cards $\implies P = \frac{12}{52} = \frac{3}{13}$.
(a) 1/26 (b) 3/13.
2
If $P(E) = 0.05$, what is the probability of 'not $E$'?
Reveal Answer & Explanation
Answer: $P(\bar{E}) = 1 - P(E) = 1 - 0.05 = 0.95$.
0.95.
3
Two dice are thrown simultaneously. What is the probability of getting a doublet (same number on both dice)?
Reveal Answer & Explanation
Answer: Favorable doublets: $(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)$ (6 outcomes). Total outcomes $= 36$. $P = \frac{6}{36} = \frac{1}{6}$.
1/6.
4
A bag contains 3 red balls and 5 black balls. A ball is drawn at random. What is the probability that it is not red?
Reveal Answer & Explanation
Answer: Total balls $= 8$. Favorable (black) $= 5$. $P(\text{not red}) = \frac{5}{8}$.
5/8.
5
Can the probability of an event be $-\frac{1}{5}$ or $1.2$? Explain.
Reveal Answer & Explanation
Answer: No, because the probability of any event is strictly bounded between $0$ and $1$ inclusive ($0 \le P(E) \le 1$). Probabilities can never be negative or exceed 1.
Probability must be between 0 and 1.
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Timed CBT Practice Tests (Exam Simulator)

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