If your newly purchased game is not showing up in your games list when you log into your Origin client, it doesn't mean you haven't purchased the rights to download it. It just means that your order is still processing.
NOTE: If you have verified that your purchase has actually gone through and processed, but still aren't able to see your games in Origin, please follow the steps below.
Also worth checking: are you on the right account? If you purchased the game online, check your e-mail inbox for the purchase confirmation e-mail, and then log into Origin with the appropriate e-mail address. You may be mistakenly using a hotmail address instead of a gmail address, or have the credentials for a different account saved up. This happens a lot.