I think 1 and 2 are correct. Then how about 3 and 4 ? 4? (space not needed)
1. A majority of people are happy with that. (OK)
2. The majority of people are happy with that. (OK)
3. A/The majority of the people is happy with that. ( Not correct, The verb should be 'are'.)
4. A/The majority of the people are happy with that. (OK)