Well, a quick look at the characters tells us there's little or no happiness for anyone in this story. Here we go:
Father-- Married to a bitter wife who never has enough--even when she gets more, it doesn't satisfy her. She claims he's unlucky. His son dies.
Mother-- Has everything--house, servants, kids, stables, clothes, and more--but it's not enough. Bitter and discontent because she feels she has no luck, despite the things she has, mentioned above. Her son dies.
Sister-- Has to also sense the throbbing financial needs of the house and the bitterness of her mother. Her brother dies.
Uncle Oscar--Perhaps a little happy, since he did make some significant money on the races; however, he also has to feel some guilt about being somewhat responsible for his nephew's death.
Bassett--The stable boy who gets Paul started on his betting "career" has made a lot of money, so he is certainly happier. However, he, too, must feel some guilt regarding the boy's death.
Paul--Hears the cries of the house and longings of his mother, understands that money is what will make her happy (because that's what she says), then discovers money is not enough. He, of course, dies trying to create more money. He tells his mother he's lucky, and there is apparently some truth to that--for awhile. He may have a moment or two of happiness in his life, like when he anonymously gifts his mother with money, but it doesn't last.
Even the house is not happy--
"And so the house came to be haunted by the unspoken phrase: There must be more money! There must be more money!"
Paul's mother says:
"If you're lucky you have money. That' s why it's better to be born lucky than rich. If you're rich, you may lose your money. But if you're lucky, you will always get more money."
Apparently, then, Paul is lucky--but he's sure not happy.
One side note to this story--some have interpreted this as a story of abuse by one or more family members against the young boy. If that happens to be your interpretation, I suppose we can say that Paul is now happy, since he is free from such torment.