\( E_i \cap E_j = \emptyset \) for \( i \neq j \) (pairwise disjoint).
\( E_1 \cup E_2 \cup E_3 \cup E_4 = S \) (cover the entire sample space).
\( P(E_i) > 0 \) for all \( i \).
Example: Any nonempty event \(E\) and its complement \(E’\) form a partition:

\[ E \cap E’ = \emptyset, \quad E \cup E’ = S \]

Law of Total Probability

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Let \(E_1, E_2, E_3, E_4\) be a partition of the sample space \(S\), where \(P(E_i) > 0\) for all \(i\), and let \(A\) be any event in \(S\).

Then, the law of total probability states that:

\[ P(A) = \sum_{i=1}^{4} P(A | E_i) P(E_i) \]

Example: Probability of Drawing a Red or White Ball

Question:

Define events:
- \(E_1\): Selecting the first bag, \(P(E_1) = 1/2\)
- \(E_2\): Selecting the second bag, \(P(E_2) = 1/2\)
Conditional probabilities:
- \(P(R | E_1) = 5/11\), \(P(W | E_1) = 6/11\)
- \(P(R | E_2) = 3/7\), \(P(W | E_2) = 4/7\)
Using the law of total probability:

\[ P(R) = (1/2 \times 5/11) + (1/2 \times 3/7) = 34/77 \] \[ P(W) = (1/2 \times 6/11) + (1/2 \times 4/7) = 43/77 \]