"The majority" and "the house" are correct in your sentences.
Singular count nouns need an article (a, an, or the), or another determiner (this, my, etc.) in front of them. "Majority" and "house" are both singular count nouns, and so need, as the choice in your two sentences, to be "the majority" and "the house."
Using articles correctly, particularly the definite article, is a skill that is difficult for English learners. In fact, even very advanced English speakers whose first language is other than English use English with the only constant imperfection being the use or non-use of "the." The problem is not only the rules; it's the numerous idioms causing problems. Nevertheless, it's a good idea to get as good an understanding of the basic usage of articles as you can.
One of the places where you can find a clear explanation of nouns and articles is in Fundamentals of English Grammar* , by Betty Azar. Chapter 11 has charts and examples that show how articles are used. The same author presents the topic of articles in a more advanced context in the highest level of her series: Understanding and Using English Grammar.**
While these and other references do help the learner to understand the rules for using articles, it takes a long time of listening and practicing to absorb them into one's being.
Attached are refereces to "Articles," a chapter in "The Grammar Book," by Marianne Celce-Murcia and Diane Larsen-Freeman. It contains suggestions for teaching articles.
*Fundamentals of English Grammar, Third Edition.. Longman. 2003
**Understanding and Using English Grammar, Third Edition. Longman. 1999