"To be bound to" stems from the meaning of the word itself, "to bind." When something is bound to something, the meaning is regarding the "sticking" of something to something. Generally, bound things are melded together so as to have a 100% chance of not coming off. If we are to extend that meaning, it would be in reference to the chance of something happening instead. If you are bound to die, that means the chance of it happening is basically 100%. Now if you think about that example, saying "have to die" or "must die" comes off very differently because they are a guarantee with an external force. Unlike "bound to," where the impetus comes innately, based on a person's disposition. "Bound to happen" is neutral in that it does not matter what someone believes or thinks (have to and must are subjective in that respect).

Ричард's answer is correct. They aren't synonyms. It means that the chance of something happening is nearly 100% or very high. Some more example sentences are: 1. You are bound to figure out the meaning of this word and how to use it, if you keep working on it. :) 2. You are bound to find a lover, if you keep searching for one on those online dating sites. 3. You are bound to find a horse trailer for cheap, if you continue to scan the Craig's List listings every morning. 4. He is bound to get fired, if he keeps showing up to work 30 minutes late every day. 5. You are bound to catch a glimpse of a humming bird at your bird feeder, if you plant red flowers nearby.