Your opponent cleared the board last turn with the help of a Flamebreak; he is playing a casual burn deck that has a lot of cards that can do four or even more damage in one shot including Ball Lightning, Rorix Bladewing and Disintegrate but also cards that do less such as Lightning Bolt. This turn he drew and cast Browbeat. Do you take the damage or let him draw the cards?
Answer: Your opponent will have one more turn after this one before he dies. If you take five damage here, he will have one draw step to find a card that can either stop Skirk Fire Marshal or kill you. I didn't tell you whether he had any ways to remove or block a creature with protection from red, because I wanted to point out that you ask: In addition to dealing damage to me, can he stop me from winning and then win later? If he can, then letting him have three cards could let him turn the entire game around. At this point, I'm going to tell you: You have not seen a card from this deck that can stop him, and have played several games. It could be there, but it is unlikely.
Your next question is exactly how many cards in his deck would constitute an out for your opponent if you took five damage. His chance of winning the game will be his chance of drawing an out. It might be quite good, but he loses if he draws mana or a low casting cost spell. The alternative is to let him draw three cards. If he does this, he now has four cards to work with: The three he draws now and the one he draws next turn. With those four cards he needs to do nine damage, and he probably has five mana this turn and eight or nine next turn to make it happen.
The exact contents of his deck are important. The fact that this is a casual game and you are facing a deck containing Disintegrate is a serious problem. Disintegrate is an out for your opponent if you take five from Browbeat, but what is worse is that next turn it can do most of the nine damage. It can't quite do the whole job, but in a deck with so many burn spells you are probably dead if he draws one: Lightning Bolt or Shock would be enough to finish you off. Rorix is a similar problem. Shock is not enough in that case since he can't bring out Rorix right now, but he has three cards and all he needs is a Lightning Bolt, Incinerate or equivalent. If he needed three spells to kill you off you would be reasonably safe. A reasonable build might include ten cards that can do half or more of the damage on their own, but with four draws to find one out of a sixty card deck he is a small favorite to find one. Once he does, he's again a favorite to find at least a Lightning Bolt type spell to finish you off.
Note that your opponent cannot win if you draw a burn spell of your own on your turn, so you ignore that possibility. In the end, letting him draw all those cards turns out to be too dangerous. Even though damage is the thing you fear the most, giving him three extra cards gives him too many ways to beat you. However, the details of this example are important and changing things a little bit has the potential to change everything. For example, what would happen if you were at seven life instead of nine?
Now one card (the card he gets on his draw phase, if you choose to take the damage from Browbeat to deny the extra card draw) is far more dangerous, as even the smallest burn spell in your opponents' deck probably kills you. Your opponent could win or tie the game off that card more than half the time if his deck is built something like we're guessing. Four cards are also more dangerous, but they look a lot more attractive as an option because there aren't that many ways he can do seven but not nine. Against the right configuration of your opponents' deck, you have a much harder decision.
Another variant: take four Mountains away from both players. In this case, instead of clearing the board with Flamebreak he used Wildfire. For the sake of argument, let's say you're currently at 7. Now how do you view Browbeat?
That changes everything. Your opponent can still kill you if you take the damage, because he only needs one spell. He still probably has the mana for that. However, getting to use four cards in time won't be easy if that's the choice you give him instead. Even assuming one of his next four cards is a Mountain he still only has one mana this turn. Next turn, he has a maximum of five and that requires him drawing two Mountains off of four cards. With so many cards in his deck effectively dead or not enough to win, you are now probably better off letting him draw three cards.
When debating whether to play Browbeat or debating putting it into a deck, you have to beware of the fact that your opponent will do his best to choose the option you do not want him to choose. If your true intentions can be disguised or your opponents are not good enough to understand their choices, cards that give your opponents options grow in power. Other cards can drastically change the relative value of different resources by presenting you with a way to exchange one for another. The classic example of that would be Necropotence. Then there are times when something that seems like it should be a good thing isn't all that good after all…
YOU (20 life): in play - Mountain x2, Blinkmoth Nexus; in hand - Mountain x2, Solemn Simulacrum, Molten Rain, Arc Slogger, Pulse of the Forge, Beacon of Destruction
OPP (19 life): in play - Urza's Power Plant (Tapped), Forest (Tapped); in hand - 6 cards; in graveyard - Sylvan Scrying
You may remember this position from my second article. By looking at the board position, it should be clear what has taken place in the previous two turns: On turn one you played Blinkmoth Nexus, on turn two you attacked with it. What other option should you have considered?
That's right, if you didn't stop to consider your options you may have already made a mistake – on the very first turn. That may seem impossible since you could not possibly have had a castable spell, but you did have one thing: access to one colorless mana. There is only one thing you can do with that mana, which is to activate Blinkmoth Nexus. That is worse than useless, since it exposes the land to removal for no reason. However, you do have one other option: You can add the mana to your mana pool and then do nothing, taking one point of mana burn. With a Pulse of the Forge in your hand, you are actually better off being at 19 than you are being at 20. That requires a lot of things to fall correctly, but in this case many of them do.
You would happily trade five or six life to be able to cast Pulse of the Forge twice, but to make that trade you need to be able to take five or six mana burn. Every point of mana burn you take is a point of mana you can't spend on something else, and there's a chance that you will never have more than one mana to spare on any of your turns. On top of that, you're going to do two damage with Molten Rain and one with Blinkmoth Nexus. If you want to reclaim Pulse of the Forge after doing those three points of damage, you'll need to mana burn yourself below thirteen. Given how this game is likely to go, it seems more likely that you will never have a chance to reuse Pulse of the Forge. Most of the time you will be putting yourself at nineteen and receive no benefits, but the risks involved are so small that the move can still be considered. You should even consider if burning for two more mana might be a better plan on turn two than attacking with Blinkmoth Nexus. If it lets you do four extra damage with Pulse of the Forge, that is more valuable to you than the damage you could have otherwise done with Blinkmoth Nexus.
This is also the mirror image of a position from my third article, where your opponent can improve his chances by taking mana burn to prevent you from reclaiming Pulse of the Fields. While in that game both life totals were important, your only damage source was Pristine Angel. If you always take four damage at a time, being at 9 is effectively the same as being at 12. For the ultimate example of wanting your life total to be as low as possible, check out old favorite Mirror Universe:
Under the old rules, you could actually kill your opponent with Mirror Universe, doing the last point of damage to yourself with a City of Brass. Under today's rules, you would die before you could switch life totals, but you don't need to abuse the rules for this card to change everything. I used to play casual decks that used Mirror Universe simply as a good card rather than to do anything abusive, and suddenly the game transforms as both players wonder what they want the life totals to be. If my opponent takes too big a lead, I'll switch the life totals and take over a commanding position. If he never does, the Mirror Universe will continue to hang over his head and he'll be no closer to winning.
Magic resources fluctuate wildly in value. There are games where the most important thing is how many lands you have in play or how many cards are left in your library. In some games life points are irrelevant. Your opponent might be playing a deck that knows it has won the game long before it is able to do even one damage to you. (His deck might not even win by damage at all.) But, over the course of typical games, life points become steadily more valuable as the players get closer to dying. On turn one, using Shock to damage your opponent trades a card for two points of damage that do not have much immediate impact on the game, but do it when your opponent is at four it can radically change the nature of the game. Do it when he's at two and he dies. Being at twenty life is valuable because it allows you to take a lot of damage without dying, often allowing you to trade those life points for extra cards or even for something as basic as the right color of mana. The danger is that as you take more damage, it becomes a bigger and bigger threat to you. The reason to avoid taking damage when you're at twenty is to avoid having to prevent it later on when you're at three. The better grasp you have of that, the better decisions you'll be making whether you're on offense or defense.