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Mutually exclusive is a statistical word defining two or more possibilities that cannot occur simultaneously. It is normally used to represent a case where the happening of one output replaces the other.

If $E$and $F$are mutally exclusive then $E鈭�F=鈭�$.

Since $E鈭�(E鈭�F)=E$and by aditivity $P(E鈭�F)=P\left(E\right)+P\left(F\right)$,

we conclude that

$P(E鈭�E鈭�F)=\frac{P(E鈭�(E鈭�F\left)\right)}{P(E鈭�F)}$

$=\frac{P\left(E\right)}{P\left(E\right)+P\left(F\right)}$.

$P(E鈭�E鈭�F)=\frac{P(E鈭�(E鈭�F\left)\right)}{P(E鈭�F)}=\frac{P\left(E\right)}{P\left(E\right)+P\left(F\right)}$

Mutually exclusive is a statistical word defining two or more possibilities that cannot occur simultaneously. It is normally used to represent a case where the happening of one output replaces the other.

Since

$E_j鈭�\underset{i=1}{\overset{鈭�}{鈭�}}{E}_i=\underset{i=1}{\overset{鈭�}{鈭�}}\left(E_j鈭�E_i\right)=\left[i鈮�j鉄�E_j鈭�E_i=鈭�\right]=E_j$

$P\left(\underset{i=1}{\overset{鈭�}{鈭�}}{E}_i\right)=[蟽-\text{aditivity}/\text{Axiom}3]=\underset{i=1}{\overset{鈭�}{鈭�}}P\left(E_i\right)$

We concluded that

$P\left(E_j鈭�\underset{i=1}{\overset{鈭�}{鈭�}}{E}_i\right)=\frac{P\left(E_j鈭�{鈭�}_{i=1}^{鈭�}{E}_i\right)}{P\left({鈭�}_{i=1}^{鈭�}{E}_i\right)}$

$=\frac{P\left(E_j\right)}{{鈭�}_{i=1}^{鈭�}P\left(E_i\right)}$.

$P\left(E_j鈭�\underset{i=1}{\overset{鈭�}{鈭�}}{E}_i\right)=\frac{P\left(E_j鈭�{鈭�}_{i=1}^{鈭�}{E}_i\right)}{P\left({鈭�}_{i=1}^{鈭�}{E}_i\right)}=\frac{P\left(E_j\right)}{{鈭�}_{i=1}^{鈭�}P\left(E_i\right)}$