I am struggling to grasp the intuition behind conditional independence. I would be very thankful if someone can help me with the below problem-
A certain baby cries if and only if she is hungry, tired, or both. Let C be the event that the baby is crying, H be the event that she is hungry, and T be the event that she is tired. Let P(C) = c; P(H) = h; and P(T) = t, where none of c; h; t are equal to 0 or 1. Let H and T be independent.
Are H and T conditionally independent given C? Explain in two ways: algebraically and with an intuitive explanation in words.
I am able to algebraically solve it and figure that the H & T are not conditionally independent but still struggling with the intuition.
Like I understand being tired(T) or hungry(H) in no way effect the probabilities of each other. Then how does the fact that given baby is crying and tired gives us additional evidence about its hunger status and vice versa?
Due to the first sentence, crying is the union of hungry and tired. Two exhaustive events are only independent when one among them covers the entirety.
Intuitively: When at least one happens, then knowing that one does not happen, means that the other must happen. Only when one among them is certain under the condition, can the two events be independent.
Mathematically: $C=H\cup T$, so $\mathsf P(H\cap T)=h+t-c$ . Hence:$$\mathsf P(H\mid H\cup T)=\dfrac{h}{c}\\\mathsf P(H\mid T \cap( H\cup T))=\dfrac{h+t-c}{t}$$
These are only equal when $(c-t)(c-h)=0$, that is that $C=T$ or $C=H$.