Class 11 Maths Probability Notes

Probability – Class 11 Maths (NCERT Based)

The chapter Probability introduces students to the mathematical study of chance and uncertainty. It is essential for understanding random experiments, statistics, and real-life decision-making.

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📖 1. Introduction to Probability

Probability measures the likelihood of an event occurring.

• Probability of an event $E$E is defined as:

$$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$P(E)=Total number of possible outcomesNumber of favorable outcomes​

• Probability always lies between 0 and 1:

$$0 \le P(E) \le 1$$0≤P(E)≤1

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🔹 2. Random Experiments and Events

1. Random Experiment: An experiment with well-defined outcomes but unpredictable results
   • Example: Tossing a coin, rolling a die
2. Sample Space (S): Set of all possible outcomes of an experiment
   • Example: Tossing a coin → $S = \{H, T\}$S={H,T}
   • Rolling a die → $S = \{1, 2, 3, 4, 5, 6\}$S={1,2,3,4,5,6}
3. Event: A subset of the sample space
   • Simple event: Single outcome (e.g., rolling a 3)
   • Compound event: Combination of outcomes (e.g., rolling an even number)

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🔹 3. Types of Probability

1. Classical Probability: Based on equally likely outcomes
   • Example: Rolling a die → $P(\text{even}) = \frac{3}{6} = \frac{1}{2}$P(even)=63​=21​
2. Empirical (Experimental) Probability: Based on experiments or observations
   • Example: Tossing a coin 100 times and getting 56 heads → $P(\text{head}) = 56/100 = 0.56$P(head)=56/100=0.56
3. Axiomatic Approach: Probability defined using mathematical rules
   • $P(S) = 1$P(S)=1, for sample space S
   • For any event E, $0 \le P(E) \le 1$0≤P(E)≤1

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🔹 4. Important Probability Rules

1. Complementary Rule:

$$P(E’) = 1 – P(E)$$P(E′)=1−P(E)

Where $E’$E′ is the complement of event E

2. Addition Rule (for mutually exclusive events):

$$P(E \cup F) = P(E) + P(F)$$P(E∪F)=P(E)+P(F)

3. General Addition Rule:

$$P(E \cup F) = P(E) + P(F) – P(E \cap F)$$P(E∪F)=P(E)+P(F)−P(E∩F)

4. Multiplication Rule (for independent events):

$$P(E \cap F) = P(E) \times P(F)$$P(E∩F)=P(E)×P(F)