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@jemoka / Jemoka Knowledge Base / wiki/concepts/mutually_exclusive.md
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--- title: "mutually exclusive" type: concept related: [Sum Rule Of Counting, Probability, Combination, Mutually Exclusive] source: https://www.jemoka.com/posts/kbhmutually_exclusive/ confidence: high status: active --- probability of “or” If its not possible for two events to happen at the same time, they are called mutually exclusive: \begin{equation} P(E\ or\ F) = P(E)+P(F) - P(E \cap F) \end{equation} This is called the inclusion exclusion principle. This is what motivates inclusion exclusion counting. General inclusion exclusion principle Its scary. Think about this: We basically need to alternate between adding and subtracting. (i.e.: in our case here, we add all the odd-group pairs (for P(x) and P(xyz)), we subtract the even-number pairs (for p(xy))). And so: \begin{equation} P(E_1\ or\ \dots\ or\ E_{n}) = \sum_{r=1}^{n} (-1)^{r+1} Y_{r} \end{equation} whereby, \(Y_{j}\) is the sum of \(P(x_n, … x_{j})\) for combinations of \(j\) events. Try not to do this.