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$$\mathbb P[C|g]=\frac{\displaystyle\mathbb P[C|G^c]\mathbb P[G^c]+\mathbb P[G|C]\sum_{k\ne g}\mathbb P[C|k]\mathbb P[k]}{\displaystyle\mathbb P[g]\left(\frac1{\mathbb P[g|C]}-\mathbb P[G|C]\right)}$$ $=\frac{\displaystyle\mathbb P[C|G^c]\mathbb P[G^c]+\mathbb P[G|C]\sum_{k\ne g}\mathbb P[C|k]\mathbb P[k]}{\displaystyle\mathbb P[C]+\mathbb P[\neg C]\frac{\mathbb P[g|\neg C]}{\mathbb P[g|C]}-P[g]\mathbb P[G|C]}$$
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Tue, 14 Sep 2021 14:17 GMT