U44 More about Probability: Essential Formulas

This section covers advanced probability concepts for DSE Mathematics, including conditional probability, the law of total probability, Bayes' theorem, and independence of events.

1 Conditional Probability

Definition and Formula

The probability of event $A$ occurring given that event $B$ has occurred is denoted by $P(A|B)$. It is defined as:

$$ P(A|B) = \frac{P(A \cap B)}{P(B)}, \quad \text{provided } P(B) > 0 $$

2 Multiplication Law & Independence

General Multiplication Law

For any two events $A$ and $B$, the probability of both occurring is:

$$ P(A \cap B) = P(A) \times P(B|A) = P(B) \times P(A|B) $$

Independent Events

Events $A$ and $B$ are independent if the occurrence of one does not affect the probability of the other. Equivalent conditions are:

$$ P(A|B) = P(A) \quad \text{or} \quad P(B|A) = P(B) \quad \text{or} \quad P(A \cap B) = P(A)P(B) $$

For three events $A$, $B$, and $C$ to be mutually independent, all the following must hold: $P(A \cap B) = P(A)P(B)$, $P(A \cap C) = P(A)P(C)$, $P(B \cap C) = P(B)P(C)$, and $P(A \cap B \cap C) = P(A)P(B)P(C)$.

3 Law of Total Probability & Bayes' Theorem

Law of Total Probability

If $B_1, B_2, \dots, B_n$ form a partition of the sample space (i.e., they are mutually exclusive and exhaustive), then for any event $A$:

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

A common special case uses two complementary events $B$ and $B'$: $P(A) = P(A|B)P(B) + P(A|B')P(B')$.

Bayes' Theorem

Bayes' Theorem relates a conditional probability to its reverse. For a partition $B_1, \dots, B_n$ and any event $A$ with $P(A)>0$:

$$ P(B_j|A) = \frac{P(A|B_j)P(B_j)}{\sum_{i=1}^{n} P(A|B_i)P(B_i)} $$

This formula is crucial for updating the probability of a hypothesis $B_j$ given observed evidence $A$.

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