# Determine the conditional probabilities

Medical investigations show that symptoms $S_1$ and $S_2$ can occur at exactly $3$ illnesses $I_1, I_2$ and $I_3$, with (conditional) probabilities $m_{i,j} = P(S_j \mid I_i)$ with $i=1,2,3$ and $j=1,2$. In matrix form they look like this: $M = (m_{ij})=\begin{pmatrix} 0.8 & 0.3\\ 0.2 & 0.9\\ 0.4 & 0.6 \end{pmatrix}$

The prior probabilities for illnesses $I_1, I_2, I_3$ are given by the vector $(0.3, 0.6, 0.1)$. Assume the illnesses exclude each other.

Determine the conditional probabilities $P(I_i \mid S_j)$, $j=1,2$, $i=1,2,3$

I think better first change $$P(I_i \mid S_j)=\frac{P(I_i \cap S_j)}{P(S_j)}= \frac{P(I_i)}{P(S_j)}$$

Now I need calculate it for every index using this formula right?

For example for $i=j=1: \frac{P(I_1)}{P(S_1)}= \frac{0.3}{0.8}= 0.375$

Next we do for $i=1, j=2: \frac{P(I_1)}{P(S_2)}= \frac{0.3}{0.3}= 1$

etc.

But not sure if this is good?

$$P (I_i \mid S_j) = \frac {P (I_i)P (S_j \mid I_i)}{\sum_{i=1}^{3}P (I_i)P (S_j \mid I_i)}$$