These words are close to each other, but nevertheless it is important to see the difference in many contexts, though there is quite a lot of overlap between the words. Let us first examine some examples:
Prone to:
1) In my childhood I was prone to colds.
2) Humans are prone to error.
Prone suggests weakness or predisposition to something regrettable and inwelcome.
Given to:
1) She is given to sudden fits of sentimentality.
2) In his recent years he is given to crying
Given to presupposes having a tendency to do something.
Apt to:
1) Being a bright child Johnny is apt to learn things quickly.
2) He is apt to understand the orders of his authorities too literally.
Apt means appropriate to the circumstances, like this is an apt decision. So apt to generally has a more positive connotation.
Closely linked words are also liable and likely:
1) Having lived
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