Probability | Class 12 Maths Chapter 13 Notes

Probability is a branch of mathematics that deals with the likelihood or chance of occurrence of an event. In Class 11, you studied the basics of probability, including random experiments, sample space, and events. In Class 12, the concept is extended to include conditional probability, multiplication theorem, independent events, and Bayes’ theorem.

Probability plays a crucial role in real-life situations such as weather forecasting, medical diagnosis, risk assessment, insurance, and decision-making.

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🔹 Key Terminology

Before diving deeper, let’s revise some important terms:

• Random Experiment: An experiment whose outcome cannot be predicted with certainty.
• Sample Space (S): The set of all possible outcomes.
• Event (E): A subset of the sample space.
• Favorable Outcomes: Outcomes that satisfy the event.
• Equally Likely Outcomes: Outcomes having the same probability.

📌 Probability Formula

$P(E) = \frac{n(E)}{n(S)}$P(E)=n(S)n(E)​

Where:

• $n(E)$n(E) = Number of favorable outcomes
• $n(S)$n(S) = Total number of outcomes

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🔹 Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred.

📌 Formula

$P(A|B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) \neq 0$P(A∣B)=P(B)P(A∩B)​,P(B)=0

📖 Explanation

• $P(A|B)$P(A∣B): Probability of A given B
• $P(A \cap B)$P(A∩B): Probability that both A and B occur

📍 Example

If we draw a card from a deck:

• Event A: Card is a king
• Event B: Card is a face card

Then:$$P(A|B) = \frac{4}{12} = \frac{1}{3}$$P(A∣B)=124​=31​

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🔹 Multiplication Theorem on Probability

The multiplication theorem helps calculate the probability of two events occurring together.

📌 Formula

$P(A \cap B) = P(A) \cdot P(B|A)$P(A∩B)=P(A)⋅P(B∣A)

$P(A)$P(A)

$P(B\mid A)$P(B∣A)

$P(A\cap B)=P(A)\cdot P(B\mid A)\approx 0.21$P(A∩B)=P(A)⋅P(B∣A)≈0.21P(A) = 0.60P(B|A) = 0.350.21

Similarly,$$P(A \cap B) = P(B) \cdot P(A|B)$$P(A∩B)=P(B)⋅P(A∣B)

📖 Explanation

This theorem is useful when events are dependent.

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🔹 Independent Events

Two events are said to be independent if the occurrence of one does not affect the occurrence of the other.

📌 Condition for Independence

$P(A \cap B) = P(A) \cdot P(B)$P(A∩B)=P(A)⋅P(B)

$P(A)$P(A)

$P(B)$P(B)

$P(A\cap B)=P(A)\cdot P(B)\approx 0.27$P(A∩B)=P(A)⋅P(B)≈0.27P(A) = 0.60P(B) = 0.450.27

📖 Key Points

• If A and B are independent:
  • $P(A|B) = P(A)$P(A∣B)=P(A)
  • $P(B|A) = P(B)$P(B∣A)=P(B)

📍 Example

Tossing two coins:

• Event A: First coin is head
• Event B: Second coin is head

These events are independent.

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🔹 Total Probability Theorem

If an event can occur in several different ways, then total probability is used.

📌 Formula

$P(A) = \sum_{i=1}^{n} P(B_i) P(A|B_i)$P(A)=∑i=1n​P(Bi​)P(A∣Bi​)

Where:

• $B_1, B_2, …, B_n$B1​,B2​,…,Bn​ are mutually exclusive and exhaustive events

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🔹 Bayes’ Theorem

Bayes’ theorem is used to find the probability of an event based on prior knowledge.

📌 Formula

$P(B_i|A) = \frac{P(B_i) P(A|B_i)}{\sum_{j=1}^{n} P(B_j) P(A|B_j)}$P(Bi​∣A)=∑j=1n​P(Bj​)P(A∣Bj​)P(Bi​)P(A∣Bi​)​

📖 Explanation

• It helps revise probabilities after new information is obtained.
• Widely used in machine learning, medical testing, and statistics.

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🔹 Random Variables

A random variable is a function that assigns numerical values to outcomes.

Types of Random Variables:

1. Discrete Random Variable
   • Takes finite or countable values
   • Example: Number of heads in coin toss
2. Continuous Random Variable
   • Takes infinite values within a range
   • Example: Height, weight

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🔹 Probability Distribution

A probability distribution gives the probabilities of all possible values of a random variable.

📌 Conditions:

• $0 \leq P(x) \leq 1$0≤P(x)≤1
• Sum of all probabilities = 1

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🔹 Mean (Expected Value) of Random Variable

📌 Formula

$$E(X) = \sum x_i P(x_i)$$E(X)=∑xi​P(xi​)

📖 Interpretation

• It represents the average outcome over many trials.

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🔹 Variance and Standard Deviation

📌 Variance

$Var(X) = E(X^2) – [E(X)]^2$Var(X)=E(X2)−[E(X)]2

$\sigma$σ

$Var(X)=\sigma^2\approx 1.96$Var(X)=σ2≈1.96μ-σ+σVar(X) ≈ 1.96

📌 Standard Deviation

$$\sigma = \sqrt{Var(X)}$$σ=Var(X)​

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🔹 Bernoulli Trials

A sequence of experiments is called Bernoulli trials if:

1. Only two outcomes: Success or Failure
2. Probability remains constant
3. Trials are independent

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🔹 Binomial Distribution

If X is the number of successes in n trials:

📌 Formula

$$P(X = x) = ^nC_x \cdot p^x \cdot (1-p)^{n-x}$$P(X=x)=nCx​⋅px⋅(1−p)n−x

Where:

• $p$p: Probability of success
• $q = 1-p$q=1−p: Probability of failure

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🔹 Properties of Binomial Distribution

• Mean = $np$np
• Variance = $npq$npq
• Standard deviation = $\sqrt{npq}$npq​

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🔹 Important Results

1. Complementary Event

$$P(A’) = 1 – P(A)$$P(A′)=1−P(A)

2. Addition Rule

• For mutually exclusive events:

$$P(A \cup B) = P(A) + P(B)$$P(A∪B)=P(A)+P(B)

• For non-mutually exclusive events:

$$P(A \cup B) = P(A) + P(B) – P(A \cap B)$$P(A∪B)=P(A)+P(B)−P(A∩B)

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🔹 Solved Examples

Example 1: Conditional Probability

A die is thrown. Find probability of getting a number >3 given it is even.

• A = {4,5,6}
• B = {2,4,6}
• A ∩ B = {4,6}

$$P(A|B) = \frac{2/6}{3/6} = \frac{2}{3}$$P(A∣B)=3/62/6​=32​

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Example 2: Independent Events

Find probability of getting 2 heads when tossing 2 coins.$$P = \frac{1}{4}$$P=41​

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Example 3: Bayes’ Theorem

Three machines produce items. Defective rates differ. Find probability item came from machine A.

(Solved using Bayes formula step-by-step in exams.)

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🔹 Applications of Probability

Probability is widely used in:

• Weather forecasting
• Artificial Intelligence
• Finance and stock markets
• Medical diagnosis
• Insurance

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🔹 Exam Tips

• Always define events clearly
• Check if events are independent or dependent
• Use correct formula (conditional vs multiplication)
• Practice Bayes theorem problems
• Revise binomial distribution thoroughly

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🔹 Common Mistakes to Avoid

• Confusing $P(A|B)$P(A∣B) with $P(B|A)$P(B∣A)
• Forgetting denominator in conditional probability
• Using independence formula incorrectly
• Ignoring total probability theorem conditions

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🔹 Summary

• Probability measures uncertainty
• Conditional probability updates likelihood
• Independent events simplify calculations
• Bayes’ theorem refines predictions
• Random variables and distributions model real-life situations

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🔹 Conclusion

Probability is one of the most practical and powerful topics in mathematics. From basic definitions to advanced theorems like Bayes’ theorem, this chapter builds a strong foundation for statistics, data science, and real-world decision-making.

Mastering this chapter requires:

• Concept clarity
• Regular practice
• Understanding formulas rather than memorizing