Can you make a living by day trading stocks?

Can you make a living by day trading stocks?-my wife takes care of my daughter so she stays home all day. would she be able to make a good amount of money as if she were to be working full time? were planning to invest $4000 and reinvest any profit, how likely is it to reach a million and how can i know if a stock will go up or down?

Since you are asking this question in this particular context, I would strongly advise you to stray away from the idea of day trading and rather counsel you to stick to investing--picking undervalued stocks across different industries while putting some money in bonds as well, depending on what market conditions are (now may be a good time to just stick with corporate bonds). 

There is no clear cut answer to how many people lose money in day trading, but I'd say that figure is around 95-98%. The few who do succeed have years--if not decades--of experience in investing/trading at large, often having read numerous books on forecasting, technical analysis and so forth (this part I'm not fully certain about, however). 

That being said, your odds of turning $4,000 into $1,000,000 by day trading stocks are slim to none. 

As for knowing how a stock will go up or down, there are contrasting schools of thought, even among technical analysts (never mind fervent value investors). It is my opinion that daily price movements under most market conditions and in mid- to large-cap stocks can be predicted with some degree of effectiveness merely through various technical analysis tools. 

I believe the most helpful are stochastics, MACD, the Elliot wave (or whatever it's called) and moving averages. Besides that, you can speculate on news (e.g. earnings releases or interest rate changes), but that's even harder to guess. 

Anyway, stay away from day trading unless you're willing to dedicate at least a year to perfecting your skills is the final advice I'll give you.
Good luck