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अध्याय 14: प्रायिकता (Probability)

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CBSE • कक्षा XI • Mathematics • अध्याय 14

अनुमानित समय: 45 Mins

प्रगति: अध्ययनरत

प्रायिकता (Probability)

In Class 11 Mathematics, "Probability" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

अध्याय CBT ऑनलाइन टेस्ट दें सभी अध्याय देखें

प्रस्तावना अध्याय अवधारणाएँ अध्याय सारांश स्व-मूल्यांकन अभ्यास टेस्ट दें

🎲 Have You Ever Wondered?

How do casinos calculate house edge down to four decimal places, or how do AI speech recognition models predict which word you are going to say next? ...

How do casinos calculate house edge down to four decimal places, or how do AI speech recognition models predict which word you are going to say next? Axiomatic probability bridges empirical outcomes with set theory.

यह अध्याय क्यों महत्वपूर्ण है

In Class 11 Mathematics, "Probability" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

अध्ययन से पूर्व (आवश्यक ज्ञान)

Theoretical probability from Class 10.
Set operations (Union, Intersection, Complement).
Permutations and combinations.

इस अध्याय के लक्ष्य

Define Random Experiment, Sample Space ($S$), and Sample Points.
Classify Events: Impossible, Sure, Simple (Elementary), Compound, Mutually Exclusive, and Exhaustive events.
State the three Kolmogorov Axioms of Probability.
Calculate probabilities using set algebra: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
Apply complementary probability: $P(A') = 1 - P(A)$ and mutually exclusive additions.

अध्याय रूपरेखा एवं प्रगति

1 1. Sample Space & Types of Events

2 2. Axiomatic Definition of Probabil...

3 3. Probability Addition Theorems

सम्पूर्ण सैद्धांतिक एवं वैचारिक अध्ययन

1. Sample Space & Types of Events

The set of all possible outcomes of a random experiment is the Sample Space ($S$):
• Tossing 2 coins: $S = \{HH, HT, TH, TT\}$.
• Mutually Exclusive Events: Two events $A$ and $B$ that cannot occur simultaneously: $\mathbf{A \cap B = \emptyset \implies P(A \cap B) = 0}$.
• Exhaustive Events: Events whose union covers the entire sample space: $\mathbf{A \cup B = S}$.

2. Axiomatic Definition of Probability

For any event $E$ of sample space $S = \{\omega_1, \omega_2, \dots, \omega_n\}$:
(1) $0 \le P(\omega_i) \le 1$ for all outcomes.
(2) $\sum_{i=1}^{n} P(\omega_i) = 1$.
(3) For any event $E$, $P(E) = \sum_{\omega \in E} P(\omega)$.

3. Probability Addition Theorems

For any two events $A$ and $B$: $$\mathbf{P(A \cup B) = P(A) + P(B) - P(A \cap B)}$$ If $A$ and $B$ are mutually exclusive: $$\mathbf{P(A \cup B) = P(A) + P(B)}$$ Complement: $\mathbf{P(A') = 1 - P(A)}$.

चित्रात्मक व्याख्या एवं मॉडल

Probability Master Matrix

Conceptual framework, core mechanisms, and analytical relationships

Academic Architecture

1. Sample Space & Types of Events • 2. Axiomatic Definition of Probability

अध्याय का सार संक्षेप एवं 10 मुख्य निष्कर्ष

मुख्य बिंदु 1

Sample Space ($S$): Universal set containing all possible random experiment outcomes.

मुख्य बिंदु 2

Mutually Exclusive: Disjoint events unable to occur concurrently ($A \cap B = \emptyset$).

मुख्य बिंदु 3

Exhaustive Events: Complete outcome set whose cumulative union forms $S$.

मुख्य बिंदु 4

Kolmogorov Axioms: Foundation establishing non-negativity and unit total probability.

मुख्य बिंदु 5

Addition Theorem: Set-theoretic formula adjusting for double-counted intersection outcomes.

स्व-मूल्यांकन अभ्यास (Check Your Understanding)

मूल वैचारिक स्पष्टता की जांच के लिए नैदानिक प्रश्न। पहले स्वयं हल करें, फिर उत्तर देखें।

A coin is tossed three times. Write the sample space and find the probability of getting at least two heads.

उत्तर एवं व्याख्या देखें

उत्तर: $S = \{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\}$ ($n(S) = 8$). Event of at least 2 heads $= \{HHH, HHT, HTH, THH\}$ (4 outcomes). Probability $= \frac{4}{8} = \frac{1}{2}$.
Sample space has 8 outcomes; P = 1/2.

If $P(A) = \frac{3}{5}$ and $P(B) = \frac{1}{5}$, find $P(A \text{ or } B)$ if $A$ and $B$ are mutually exclusive events.

उत्तर एवं व्याख्या देखें

उत्तर: For mutually exclusive events, $P(A \cap B) = 0$. $P(A \cup B) = P(A) + P(B) = \frac{3}{5} + \frac{1}{5} = \frac{4}{5}$.
4/5.

If $P(A) = 0.5$, $P(B) = 0.3$, and $P(A \cap B) = 0.2$, find: (i) $P(A \cup B)$, (ii) $P(A')$.

उत्तर एवं व्याख्या देखें

उत्तर: (i) $P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.5 + 0.3 - 0.2 = 0.6$. (ii) $P(A') = 1 - P(A) = 1 - 0.5 = 0.5$.
(i) 0.6; (ii) 0.5.

Two dice are thrown simultaneously. What is the probability of getting a sum greater than 9?

उत्तर एवं व्याख्या देखें

उत्तर: Total outcomes $= 36$. Favorable outcomes with sum $> 9$ (sums 10, 11, 12): $(4,6), (5,5), (6,4), (5,6), (6,5), (6,6)$ (6 pairs). $P = \frac{6}{36} = \frac{1}{6}$.
1/6.

Explain why $P(A \cup B) = P(A) + P(B)$ is true only for mutually exclusive events.

उत्तर एवं व्याख्या देखें

उत्तर: Because for mutually exclusive events, the intersection $A \cap B = \emptyset$, so $P(A \cap B) = 0$. When events overlap, the intersection probability must be subtracted to prevent double-counting shared elements.
Shared intersection is empty; otherwise double-counting occurs.

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