Mining rewards are paid to the miner who discovers a solution to the puzzle first, and the probability that a participant will be the one to discover the solution is equal to the portion of the total mining power on the network. Participants with a small percentage of the mining power stand a very small chance of discovering the next block on their own. For instance, a mining card that one could purchase for a couple of thousand dollars would represent less than 0.001% of the network's mining power. With such a small chance at finding the next block, it could be a long time before that miner finds a block, and the difficulty going up makes things even worse. The miner may never recoup their investment. The answer to this problem is mining pools. Mining pools are operated by third parties and coordinate groups of miners. By working together in a pool and sharing the payouts among all participants, miners can get a steady flow of bitcoin starting the day they activate their miner. Statistics on some of the mining pools can be seen on Blockchain.info.
"I've done the math. Forget mining. Is there a less onerous way to profit from cryptocurrencies?"
As mentioned above, the easiest way to acquire bitcoin is to simply buy it on one of the many exchanges. Alternately, you can always leverage the "pickaxe strategy." This is based on the old saw that during the 1849 California gold rush, the smart investment was not to pan for gold, but rather to make the pickaxes used for mining. Or, to put it in modern terms, invest in the companies that manufacture those pickaxes. In a cryptocurrency context, the pickaxe equivalent would be a company that manufactures equipment used for Bitcoin mining. You may consider looking into companies that make ASICs equipment or GPUs instead, for example.