What is the proper manner in which to thank a supervisor for allowing you to take a really long shot? I am talking about when a professor (supervisor) who did not really know you decides to give you a chance, even though there is clearly a very low probability of success. Especially in the case where the supervisor invested significant physical resources and/or their time and you still failed to accomplish the goal (in this case research result). I am looking for a formal and appropriate way to convey my gratitude to the professor for taking the risk. Again this is in the context where you have specifically failed to reach the goal and cost the supervisor in question a good deal of resources in the process. Is it even proper to convey gratitude at this juncture for giving you an outside shot at tremendous opportunity? Or should I just be extremely apologetic?