I have a list of sentences and each sentence is classified with a number of emotions ex:
I loved the movie Happiness $= 1$ Disappointment $= 0$
I hated the move Happiness $= 0$ Disappointment $= 1$
A great movie, Good movie Happiness $= 1$ Disappointment $= 0$
Poor acting Happiness $= 0$ Disappointment $= 1$
Great acting but poor story line Happiness $= 1$ Disappointment $= 1$
I already found the probability for each of these:
$$P(\text{Happiness}) = \frac{3}{5}$$
$$P(\text{Not Happiness}) = \frac{2}{5}$$
$$P(\text{Dis}) = \frac{3}{5}$$
$$P(\text{Not Dis}) = \frac{2}{5}$$
Now I would like to try to find:
1) $P(\text{Hap|Dis})$
2) $P(\text{Hap|Not Dis})$
3) $P(\text{Not Hap | Dis})$
4) $P(\text{Not Hap | Not Dis})$
I calculated the prob of the above manually as:
1) $\frac{1}{5}$
2) $\frac{2}{5}$
3) $\frac{2}{5}$
4) $\frac{0}{5}$
I guess if I need to find $$P(\text{Hap|Dis}) = \frac{P(\text{Dis|Hap}) \cdot P(\text{Hap})}{ P(\text{Dis})}$$
I know that $$P(A|B) = \frac{P(A \cap B)}{P(B)}$$
so $$P(\text{Hap|Dis}) = \frac{P(\text{Hap } \cap \text{ Dis})}{P(\text{Dis})}$$ and that $$P(A \cap B) = P(A)\cdot P(B)$$
so $$P(\text{Hap|Dis}) = \frac{P(\text{Hap})\cdot P(\text{Dis})}{P(\text{Dis})}$$
So $$P(\text{Hap|Dis}) = \frac{\frac{3}{5} \cdot \frac{3}{5}}{\frac{3}{5}}$$ Why is the answer not $\frac{1}{5}$? Am I missing something pls?
The problem here is -
Consider the expression
$$P(A∩B)=P(A)⋅P(B)$$
This is valid when $A$ and $B$ are independent events, which clearly is not the case here.
Hope this answered your question.