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Cheat sheet · Class 12 · CBSE · Commerce Mathematics

Probability cheat sheet

The whole chapter on one page: key formulas, facts and the points to remember, for quick revision before a test.

CBSE Class 12 · Mathematics

Cheat sheet

Chapter 13Probability

Bayes: reverse the condition — P(cause | effect).

1. Conditional Probability

P(A∣B)=P(A∩B)P(B)P(A|B) = \dfrac{P(A\cap B)}{P(B)}
P(A∩B)=P(A)P(B∣A)P(A\cap B) = P(A)P(B|A)

Independent: P(A ∩ B) = P(A)P(B).

2. Total Probability & Bayes

P(A)=∑P(Ei)P(A∣Ei)P(A) = \sum P(E_i)P(A|E_i)
P(Ei∣A)=P(Ei)P(A∣Ei)∑jP(Ej)P(A∣Ej)P(E_i|A) = \dfrac{P(E_i)P(A|E_i)}{\sum_j P(E_j)P(A|E_j)}

Bag I: 3 red, 4 black; bag II: 5 red, 6 black. A bag is picked at random and a red ball drawn. P(bag I | red) = (½·3/7) ÷ (½·3/7 + ½·5/11) = 33/68.

3. Random Variable

E(X)=∑xipiE(X) = \sum x_ip_i
Var(X)=E(X2)−[E(X)]2\text{Var}(X) = E(X^2) - [E(X)]^2

4. Key Points

  • Mutually exclusive ≠ independent.
  • Sum of probabilities in a distribution = 1.
  • Partition: events disjoint and exhaustive.
  • Draw a tree diagram for multi-stage problems.

Remember: P(A|B) | multiplication | total probability | Bayes | mean

Probability: questions and notes

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