Hifumi's Study NotesnotesProbability and StatisticsAdditive Rule

Additive Rule

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PAS

Cegep/2

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The cardinality of an event that can occur through $k$ events is

For mutually exclusive events, $n (A_{1} \cup A_{2} \cup \dots A_{k}) = n (A_{1}) + n (A_{2}) \dots + n (A_{k})$
For non mutually exclusive events, $n (A_{1} \cup A_{2} \cup \dots A_{k}) = \sum_{i = 1}^{k} n (A_{i}) - \sum_{i = 1}^{k} \sum_{j = i + 1}^{k} n (A_{i} \cap A_{j}) + \dots + (- 1)^{k + 1} n (A_{1} \cap A_{2} \cap \dots A_{k})$ (aka. inclusion-exclusion principle)

Additive rule of probability

The probability of an event that can occur through $k$ events is

For mutually exclusive events, $P (A_{1} \cup A_{2} \cup \dots A_{k}) = P (A_{1}) + P (A_{2}) + \dots P (A_{k})$
For non mutually exclusive events, $P (A_{1} \cup A_{2} \cup \dots A_{k}) = \sum_{i = 1}^{k} P (A_{i}) - \sum_{i = 1}^{k} \sum_{j = i + 1}^{k} P (A_{i} \cap A_{j}) + \dots + (- 1)^{k + 1} P (A_{1} \cap A_{2} \cap \dots A_{k})$ (aka. inclusion-exclusion principle)

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