might do well = there is a possibility: - She might do well in the advanced class. may as well = why not? what do I have to lose? what difference will it make?: - I'm already wet, I may as well keep walking in the rain. - The band didn't show up and nobody I know is here, I may as well go home now.

Are you thinking of 'might do well' in a phrase such as 'You might do well to change your car insurance provider' ? If so, it's a phrase used when giving advice. This is different from 'might as well'. 'Might as well' is often said with a literal or figurative shrugging of the shoulders, meaning 'It would be the same'. For example, if you are at the back of a long queue, and it's obvious that they'll have run out of tickets by the time you get to the front of the queue, you could say 'We might as well go home', meaning there's no point in staying. Whether you go home now or wait to be told that there are no tickets, the result will be the same - no tickets. Here's another example. There's something that you have been trying to keep secret, but you realise that this will soon no longer be a secret. In this situation, you could say to your friend, 'I might as well tell you this now, because soon everyone's going to know....'. This means that whether you tell your friend now or wait for him/her to hear it from somebody else, the result will be the same - he/she is going to find out about your secret. So, going back to the original example, if you say 'You might do well to change your car insurance provider' it means that I think you ought to do this. Perhaps I think that they are poor value and you should change. Someone could say 'You might as well change your car insurance provider' if you are planning to change your home and health insurance provider, and you think it would do no harm to change the car insurance provider as well. The implication here as that the two providers offer the same service but it would simply be easier to have all your insurance with one company. Here 'might as well' means 'Why not?'