13Class 12 Maths

Probability formulas

Conditional probability, Bayes' theorem and random variable formulas for Class 12. Writing what's given in P(A|B) notation solves half of each problem.

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All Maths formulas

Conditional probability

Conditional probability: $\displaystyle P(E|F) = \frac{P(E \cap F)}{P(F)}, \quad P(F) \ne 0$
Multiplication rule: $\displaystyle P(E \cap F) = P(E)P(F|E) = P(F)P(E|F)$
Independent events: $\displaystyle P(E \cap F) = P(E)P(F)$

Total probability and Bayes

Total probability: $\displaystyle P(A) = \sum_i P(E_i)P(A|E_i)$
The Eᵢ partition the sample space.
Bayes' theorem: $\displaystyle P(E_i|A) = \frac{P(E_i)P(A|E_i)}{\sum_j P(E_j)P(A|E_j)}$

Random variables

Mean: $\displaystyle E(X) = \sum x_iP(x_i)$
Variance: $\displaystyle \text{Var}(X) = E(X^2) - [E(X)]^2$
Binomial probability: $\displaystyle P(X = r) = {}^nC_r\,p^rq^{n - r}, \quad q = 1 - p$
Binomial mean and variance: $\displaystyle \text{mean} = np, \quad \text{variance} = npq$

Where students slip up

Mixing up P(A|B) and P(B|A). Bayes' theorem exists because they're different.
Treating independent and mutually exclusive as the same thing.
Forgetting that probabilities in a distribution must add up to 1.

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