From Right Field: Probabilities, Randomness, and Averages for Non-Engineer Magic Players
Over the past few weeks my columns have included additional sections that tended to be about the laws of randomness and probability what laymen ("Heh he said 'lay' and 'men.'") call the law of averages. A lot of people seem to be confused about these topics. Others just wanted to know more. I've typically not gotten involved in these types of detailed columns about complicated theories that seem to exist in the Magic universe. For instance the series of articles on card advantage went way over my head when it got past the idea of drawing cards = good stuff.
(Digression The First: What most people call"The Law of Averages" is actually called"The Law of Large Numbers." For the sake of more efficient communication I will continue to use the phrase"The Law of Averages.")
It was suggested by a reader (okay it was Heidi Klum) that given my background I could maybe you know do a little something that concerning Magic and probabilities that was both entertaining and accessible. You don't say no to Heidi.
Coins - The Stats 101 Professor's White Meat
Averages tell us only the likelihood of a certain outcome of an event. It doesn't tell us exactly what outcome will occur the next time the event happens (unless the percentage is one hundred or zero). In statistics-speak The Law of Averages says that"in repeated independent trials with the same probability p of success in each trial the chance that the percentage of successes differs from the probability p by more than a fixed positive amount e > 0 converges to zero as the number of trials n goes to infinity for every positive e." (Prof. Phillip B. Stark U.C. Berkley)
I know. I said this was for the less geeky geeks. So once again then but in English this time. The classic examples involve flipping a coin. Typically we say that there is a 50% chance of the coin landing heads up. That leaves a 50% chance that it will land tails up. That definition in the previous paragraph means that means that the more you flip the coin the more likely it is that the total of the heads or tails will be half of the total flips. Flip it just six times and there is a chance that you will see all heads or all tails. Not a very good chance but a chance nonetheless. Flip the coin 100000 or 100000000 times and you'll be much closer to 50/50. Flip it an infinite number of times and you'll see 50/50. You'll also be very very old.
Unfortunately as with almost any example that anyone has ever given involving anything we have to make certain assumptions.
Coin Flipping Assumption #1 - The coin will not land on its side during any number of infinite flips: Seems kinda silly doesn't it? Heck it's so improbable that it was worked into an episode of The Twilight Zone. Yet it could happen. When discussing the simpler aspects of probabilities it's always best if the numbers are nice and round. So we're going to have to pretend that the coin will never land on its side. (And before you ask yes I've seen it happen. Freaked me out like I'd found out that Crispin Glover was dating Jessica Biel.) [Romeo is providing his own cheesecake this week and doing a fine job. - Knut]
Coin Flipping Assumption #2 - The coin is aerodynamically perfect and will not favor either side: Again this might seem silly but it has to be taken into account. For example because of the design of Lincoln's bust on the front and the memorial on the back a penny flipped into the air and allowed to come to rest of its own volition will actually land heads up more than 50% of the time. It's a minute percentage of the time along the lines of 50.1% but it means that you have to remove that from the discussion. (It also means that if your opponent gives you the choice of making the call when flipping a penny you should call heads.)
Okay so finally we get to flip that coin. Woo hoo! Go ahead. Flip it. It came up heads. Mark that down.
Now given the fact that a coin has a one-in-two chance of coming up tails and it just came up heads what is the probability that it will be tails on the next toss?
Think about it.
Okay you're a bunch of smart cookies. You knew that there was still only a 50% chance of being tails on the next flip. Hey you'd be surprised by how many people will misinterpret probabilities and say that the next flip is going to be tails 100% of the time. You can understand their logic."It's supposed to be tail once every two times. It was just heads. This next one has to be tails." Faulty logic but logic nonetheless. [For more coin-flipping as art see Tom Stoppard's incredible"Rosencrantz and Guildenstern Are Dead" or the movie of the same name. - Knut]
Independent and Dependant Events
The reason that the second flip is not guaranteed to be tails is simply that the first flip has no bearing on the second flip. In statistics parlance the two events are completely independent. When events are independent of each other the probabilities for a particular outcome of a future event don't change based on the outcome of previous event. If you flip a coin twenty times and it comes up heads the first nineteen times there's still just a 50% chance that the twentieth flip will produce a result of tails.
"Wait a second. Did you just say nineteen a row? How likely is that?"
Okay now the events are dependant. The only way to have nineteen heads in a row is to have eighteen first. The only way to have eighteen in a row is to have seventeen. And so on. The chances of the first one being heads is one in two. That's the same as the second through nineteenth times. The chances of it being heads on both the first two flips is figured by multiplying the two together. That's 1/2 times 1/2 which is 1/4 or 25%. To find out what it would be for nineteen flips we throw in seventeen more of those 1/2 suckers. That brings us to 1/524288. (Someone check my math on that um-kay?) In other words there is a one in 524288 chance of flipping a coin nineteen times and having it come up heads (or tails) every time. So it could happen. It's just not bloody likely in your lifetime. Well not unless your whole career is geared around coin flipping.
Nineteen in a row though? If there was some sort of wager on the outcome of those nineteen coin flips though I'd think there might have been some cheating going on. Hey that's just me. (Oh come on you knew I was going there at some point.)
Playing Cards - The Stats 101 Professor's Other White Meat
Now we're getting closer to Magic. For tougher probability questions statistics professors move on to playing cards. So we will too. First though we have to take care of an assumption or two.
Card Drawing Assumption #1: The deck is"sufficiently randomized" - That phrase keeps coming up over and over again in articles and forums relating to Magic. I'm not getting into that phrase as it relates to a Magic deck. All it means in this case is that the deck was shuffled well enough that it the cards are in a random order. In other words a card that was on top during one draw can be in any position in subsequent draws made from a deck of the same cards.
Card Drawing Assumption #2: No card is more likely to be in one particular place in the deck than in another - this is more like a corollary to the first one. As you know in Magic decks lands often stick together because they are handled more than other cards. They get coated in dirt and oils that cause them to stick to each other. None of that is going on in these examples.
"Oooooo . . . drawing cards."
Take a deck of regular playing cards. Remove any Jokers and twelve other cards making sure that you've left the Jack of Spades in the deck. You have forty cards. Shuffle 'em up. What is the chance of you drawing that Jack of Spades from the top of the deck? It's one in forty.
Draw a card. Ta da! It was the Jack of Spades. Were you"lucky" or what?
The answer is"or what." There's no such thing as"luck." I've covered that before. You had a 2.5% chance of getting the Jack of Spades. You got the Jack of Spades. Randomness. On average you were going to get the Jack of Spades once every forty times. This was one of those times. No magic. No outside forces controlling your draw. Simple randomness.
We also know that this doesn't mean that you won't get the Jack of Spades the next thirty-nine times you draw from the same forty-card deck (i.e. you're putting the JoS back in before shuffling). Just like the coin flips the next time you draw from these same forty cards is independent of the last and all previous draws.
Now put the other twelve cards back keeping the Jack of Spades aside. Sufficiently randomize the deck. (Shaddup!) Remove the top twelve cards of the deck. Shuffle the Jack of Spades back in. You still have a one-in-forty chance of drawing the Jack of Spades from the top of the deck.
(Digression The Second: What we're simulating here is a certain point in two different games of Magic in which there are forty cards left in your deck and one of them is the same specific card in both cases.)
Draw a card. It was the Jack of Spades again? What it going on here? Have we violated the laws of probability?
No. The laws of probability are like the laws of gravity. You can't violate them. As a professor of mine used to say"A law is a law because you can't change it." (This was in engineering school not law school. The whole point of law school of course is learning how to change laws. Or get around them.)
Of course what you can do is change the situation. This doesn't violate the laws of probability. It changes the probability. For example let's say that you keep the Jack of Spades aside. You shuffle the other thirty-nine cards. Put the Jack of Spades on top.
Your chances of randomly drawing the Jack of Spades remain one in forty. However we don't draw randomly in Magic do we? We draw from the top of the deck. Go ahead. Take that card on top. It was the Jack of Spades right? Because of the fact that we changed the situation by putting the JoS on top of the deck your chance of drawing it from the top of the deck was 100%.
Quickly Relate this to Magic Before I Fall Asleep
Your chance of drawing any particular card in your Magic deck at a particular point in the game is very easy to figure. How many of that card are left in your deck? How many total cards are left in your deck? Divide the first number by the second number. Move the decimal two places to the right. That's the percentage chance you have of drawing a copy of that card at that time.
(Note: We're not talking about the chance that you have to draw a particular card throughout the course of the game. That is dependent on too many factors (e.g. how many cards are drawn in total) to deal with in this piece. All we're looking at is what the chances are of drawing a certain card right now.)
Let's take a simple example. You're pretty deep into the game. Your deck has punked out on you. You have no cards in hand and no permanents like Serum Tank that help you draw cards. You have had nothing like Vampiric Tutor to put any particular card on the top of your deck. In other words you are completely dependant on your draw and you have no clue what that card might be. Worse your opponent will kill you if the game gets to her turn. She doesn't even need to draw. She doesn't need any of the cards in her hand. What's on the board will win it for her.
You have forty cards left in your deck. Only one of them can save you from certain doom. In fact if you get that card you actually win the game. If you don't you lose. It's that simple. (For the rest of this piece for convenience sake this will be known as The One-in-Forty Spot.)
What are the chances that you win the game?
"Hold on a second Romeo!"
"That is such a bad example. I need to know more. Why am I going to lose? What does my opponent have on the board? What cards have we played? That information changes everything. Without knowing how I got there I can't answer that. Give it up. You're so bad at this."
Actually I'm really very good at this. I even get paid to do it. Working for an insurance company requires doing a lot of these probability exercises. (You don't even want to know how many variables the actuaries use to compute something as simple as the cost for collision coverage for a car. The matrices are mind-numbingly boring. That's why they get the big bucks.)
This example is simply what it is. I know that a lot of you want to know what the board position is. A lot of you want to know what cards have been played. Maybe it's because you need concrete examples. (I'm like that. Abstracts bother me. I need to be able to visualize the situation.) Maybe it's because you want to try to poke holes in the example rather than dealing with it.
For whatever reason you want to know the particulars the answers to those questions just don't matter. The game is in the state described above. Make up what you want. Maybe your opponent has a thousand Insect tokens ready to swarm while you're at 999 life with only one blocker and nothing else. Maybe you're at one life and your opponent has a Soldier in play while you have no non-land permanents on the board. Whatever makes you happy.
Simply put we're talking about those darn independent events again. While it might be interesting to figure out how the game got into the state it's in it doesn't matter for the particular probability we're talking about. The bottom line is this: you need the top card of your deck to be one particular card (of which you only have one copy left in your deck) or you lose. Period. What happened before does not change the fact that you are interested in only one of forty cards left in your deck.
(If you are one of the people who think that yes one or more of those answers do actually matter and thus you will regard anything written after this as bogus you might as well leave now. I won't be able to convince you no matter what I write. Use the time you would have spent reading the rest of this to buy some StarCity Angel tokens or something.)
None of the specifics of how you got here matter from a probabilities perspective. What matters is that your game is completely dependant on getting that one card in your deck.
What are your chances of getting that card? One in forty. Two point five percent.
You draw the card. You got it. Are you"lucky"? No. Again there's no such thing as"luck." That card was on top of your library as a result of random chance. Just as with the Jack of Spades we would expect this to happen once every forty draws. This was one.
Two matches later you find yourself in the same One-in-Forty Spot. Whether it's the same card you need or not doesn't matter. You are again in a situation in which only one card of the forty left in your deck can keep you in the game.
(Digression The Third: Of course from a Magic perspective how you got here does matter. I mean why are you playing a deck that does this to you? Why do you only have one answer to all of this stuff and no way to draw cards? And why do you keep playing yourself into a corner like this? Those however are issues for a strategy piece. This is not that piece. Suffice it to say you should be playing a better deck.)
You draw your card. Lo' and behold you got that one card again.
Given that the chance of doing this twice in a row is one in 1600 you can understand why people might believe in"luck." However in this example it was just random chance. A one-in-1600 chance is still a chance. It was bound to happen at some point to someone. It happened to you twice in a row in the same circumstance. Congrats.
You see averages and odds don't tell us what will happen the next time a random event occurs. They only tell us what we can expect if the event were to occur a sufficiently large number of times. (See Prof. Stark's definition at the beginning of this piece.) Because of randomness there is no way to tell what the next coin flip or card drawn will be. We just know what we should expect. Sometimes unlikely events occur back to back or get bunched up. It happens. That's randomness.
On the flip side (horrendous pun intended) just because unlikely events can indeed occur back to back or several times in a small sample doesn't mean you should turn a blind eye when they do happen. Events that consistently occur against the odds are a great indicator that"something is up" so to speak. You owe it to yourself to investigate what's going on. You might learn something.
The Cancer Clump
I'll give you a real life example from right here in Knoxville. In the early 1990's there was a neighborhood in town in which an inordinate number of people were developing cancer. Doctors from the University of Tennessee said it was five to seven times what they would have expected to see. It wasn't one type of cancer though. It wasn't several types that might be related to the same carcinogen either. It was simply a much higher percentage of people than would be expected getting cancer over a three or four year period.
Of course given how high a percentage of people were developing cancer it was investigated. The city investigated. Doctors from U.T. investigated. Some of the homeowners hired water and air specialists as well as people to look into the houses themselves.
None of them could find anything that was causing the unusually high rate of cancers. It wasn't anything in the air. It wasn't anything in the water. It wasn't asbestos or radon or anything else in the homes. It wasn't related to where they worked what they ate or where they went to have fun. It wasn't that they were all related (i.e. genetic). There was nothing at all to connect all of these cases except that these people lived in the same neighborhood.
The conclusion was simple. There were just a lot more people in that area getting cancer in those three or four years than would be expected. That was it. (Since then it's been back to normal.)
This didn't ease the fears of the people who lived there of course. When you see something like that happening at such a high rate you hope you can find an answer. You want someone or something to be responsible. You want to know what you can take precautions against and who to blame. When it's just a clump when it's just randomness well it's scary.
It's completely understandable how those folks felt. They saw something like cancer sweeping over them like the plague. When someone finally realized that this was happening they investigated. It would have been irresponsible not to investigate. What if there had been something in the air or water something that made living in that area unhealthy? Those homeowners owed it to themselves their children and people in other neighborhoods to sound the alarm. It doesn't matter that it turned out to have no cause that anyone could find. They did the right thing in saying"Hey cancer is happening here at a much higher rate than normal. Can anyone tell us why?"
When we see unlikely events occur at a significantly higher rate than expected we should investigate to see if there is a reason. We'll save the rest of this rant for a bit later though.
Magic: The Randomness
As I (and many others) have stated before Magic is game of randomness. As far as game theories go there are several that apply to Magic. It's a game of incomplete information. You usually don't know what your opponent is holding or what cards are where in either of your decks. It's also a game of resource management. The way that you manage your resources (e.g. cards in hand life total permanents) helps determine the outcome of the game. (Your opponent also has something to do with the outcome. Usually. Heh.)
At its heart though it's a game of randomness. After all it's a card game. Cards are shuffled and dealt. We draw cards off of the top of the deck (unless something else says to do otherwise). We help ourselves win by minimizing this randomness. By minimizing the randomness we change the probabilities.
The Rule of Four
The first way players usually learn to minimize randomness is by having more than one copy of a card in our deck. Let's take a Goblin deck from the current Standard environment for example. We're allowed by the rules to have four copies of Goblin Warchief in them. Because the Warchief makes the player's Goblins cost less and have haste a Goblin player will usually play the maximum allowed (four) assuming he or she can get his or her hands on them. Since they're uncommon (read: cheap and easy to find) that's not tough.
Now let's say that a player has four distinctly different Goblin Warchiefs. One is a boring old English copy. One's a foil. One's in Chinese while the fourth is in German. Let's go back to our One-in-Forty Spot.
For whatever reason your next card just has to be a Goblin Warchief. You have quite against the odds not seen one all game but that's beside the point. All that matters is that given your board position and what's in your hand a Warchief off the top of your deck wins the game. If you draw any card that's not the Warchief and you lose.
What's your chance of drawing the foil Goblin Warchief? It's one in forty.
Ah there's the beauty of the Rule of Four. You don't care which Warchief you get although it would be keen what with the style points and all to win with the foil Warchief. You'll take any Goblin Warchief you can get though.
What is your chance of drawing any of the Warchiefs? Since you have four copies left in your deck you have a 10% chance. By having four copies in your deck you increase your chances of getting a Warchief fourfold or 400%.
Topdecking: Not a Skill You Can Practice / Not a Way to Reduce Randomness
Topdecking is when a desired or needed card is randomly drawn off the top of the deck during the player's regular draw of the turn (i.e. during the draw step as allowed by Rule 304). Topdecking then is simply the result of random chance. If you get a card that you want or need as a result of your regular draw you've topdecked. If not you haven't. End of story.
Topdecking is not a way to reduce randomness. Topdecking just happens and in fact is a result of random chance just as the above definition says.
Here are some things that are not topdecking:
Drawing the card you need when you have just played Vampiric Tutor: There's nothing random here. Vampiric Tutor puts the card you want on top of your library.
Getting the card you need out of the ten you just drew off of Rush of Knowledge (with Broodstar on board): This isn't topdecking because it didn't happen as a result of the Rule 304 draw. It happened because you drew somewhere between 16.67% and 100% of your friggin' deck.
Getting the card you need with a Wish of some sort: Again nothing random here.
Getting the card you need by cheating: Yet again nothing random. Also it's against the rules.
Because topdecking is the result of random chance the probability of topdecking a card is the same probability as randomly drawing that card. However the more"good" cards (i.e. cards that help you in the given situation at the time of the draw) you have in your deck the more likely you are to topdeck.
Funny how that works huh?
(Digression The Fourth: This is why some of the best decks produce so many comments along the lines of"He topdecked again!" When you have eight or ten or fifteen ways in your deck of dealing with something you're that much more likely to draw a card that can help you.)
Silver Knight + Worship = Red Deck Loses 2k4
Let's say that I'm playing my Goblin deck with a hint of Green. I do that so that I can bring in as many as four Naturalizes and four Nantuko Vigilantes from the sideboard. Unfortunately I am facing a White Weenie deck that has a Silver Knight and a Worship on board. I have a slew of Goblins. My opponent is at one life and none of it matters if I can't get something to deal with Worship. With me at two if I don't draw Naturalize or Nantuko Vigilante all he has to do is swing through the Goblins and I'm done. His hand is empty so I don't even need to worry about something as silly as Razor Barrier protecting the Worship. (Sadly I have no Goblin Taskmasters in the deck either. So I can't even make a colorless face-down morph to stem the bleeding this turn. I know that the Vigilante is a morph. If I get that card though I'm not using it for blocking a Silver Knight. I'm destroying the Worship so that I can win. If only I'd listened to that good-looking writer Chris Romeo and played Goblin Replicas and Pyrite Spellbombs in my deck!) [You'd also be playing *ahem* Goblin Sky Raider. - Knut keeping Romeo's ego in check]
I have a certain number of cards left in my deck. Since I seem to like forty we'll say that I have forty left. I haven't seen a Naturalize or a Vigilante all game and I brought in the complete set of four each I know that I have a one in five chance of drawing something that deals with Worship. If I draw one it's still topdecking. However by giving myself such good odds at drawing a spell that can deal with an enchantment I make it more likely to pull off a so-called topdeck.
The Department of Redundancy Department
Redundancy occurs when you have more than one copy of a card or an effect in your deck. I don't think I've ever seen this stuff defined. So in an attempt to make my mark on the Magic Theory Community I'm going to create a couple of definitions.
There are two types of redundancy. First there is absolute redundancy. Absolute redundancy occurs when there is more than one copy of a single card in the deck. In the previous example we have an absolute redundancy of three as it relates to Naturalize since there are four copies of it. Ditto with Nantuko Vigilante.
The second type of redundancy is effective redundancy. Effective redundancy occurs when you have duplicates of a certain effect in the deck. (This also gives rise to the fact that all absolute redundancies are effective redundancies but the opposite is not always true.) Using the same example we have an effective redundancy of seven as it relates to destroying artifacts and/or enchantments. Since we have four copies of Naturalize and four copies of Nantuko Vigilante we have eight ways to destroy an artifact or enchantment. In other words we effectively have eight Naturalizes in the deck. (We can't say that we effectively have eight Vigilantes however when looking at Naturalize and the Vigilante together because we can never swing with Naturalize.)
Redundancy allows you to load up on an effect. In the example above I was almost certain that my opponent would bring in Worship since Worship plus Silver Knight is sure lock against most Red decks (which is also why Red decks absolutely must run Pyrite Spellbomb). By bringing in eight cards that could deal with Worship I was able to change the probability that I could deal with Worship. Heck I could bring in four Creeping Molds too. Then instead of eight out of sixty cards (13.33%) being able to deal with Worship there would be twelve or 20% of my deck that could kill Worship. I wouldn't do that though 'cause that would just be silly.
(Digression The Fifth: Disregard the fact that 8/15ths of my sideboard are dedicated to artifact and enchantment destruction. This would leave me only seven cards to deal with any other types of problems. Let's just say either that (1) the deck deals incredibly well with almost everything other than Worship and Affinity or (2) I just don't care about winning those other matchups.)
"On Three Draw!"
Our favorite way of reducing the effects of randomness is with card drawing. The Rule of Four which encourages absolute redundancy is so ingrained in us that we don't get any special joy or fun out of adding four copies of most cards to our decks. Oh but drawing cards. Man oh man do we like that. There is just no feeling like casting Rush of Knowledge with a Myr Enforcer on the board and mana left to counter something if we need to. A fistful of cards is like a bag full of Halloween candy to a nine-year-old kid. Or a roomful of nine-year-old kids to Michael Jackson.
(from Esquire magazine)
(Q: What did Michael Jackson say when he met Woody Allen?)
(A:"Do you have two fives for a ten?")
The subject of card drawing has been done to death and by better players and theorists than me. So I'll say it the way Nuke LaLoosh might have said it in Bull Durham if Bull Durham had been about Magic:"Woo-hoo! I love drawing cards! It's like you know better than not drawing cards."
(Digression The Sixth: I am not going to address tutoring because those effects put a particular card in a certain place. While probabilities tell us the chance of getting the card that creates the tutor effect once the effect is final we have a near 100% chance of getting the desired card. Again this is as it regards tutoring and not consultation effects like Tainted Pact.)
(Digression The Seventh: I say we have a near 100% chance of getting what we want with tutor effects because we could still do something stupid. The classic example - and if someone could point me to it I'd love to read it again - is from the game in which a player played Vampiric Tutor on his own turn despite the fact that (a) it's an instant and (2) his opponent had a Rootwater Thief on the board. The opponent swung with the Thief activated the ability and without looking at his opponent's deck said"I'll take whatever you put on top of your library with the Tutor.")
Mana from Heaven
The most overlooked way that we can increase our chances of getting and being able to play a certain card is to make sure that our mana base is as the kids say"tight." For example true weenie decks don't need twenty-four lands. Their spells often don't get more expensive than two or three mana. Because of this they're able to run fewer lands. Each slot that isn't a land is obviously a spell. A weenie deck that runs only eighteen lands has six more spells than a deck that runs twenty-four lands. On the other hand a deck that has a lot of expensive spells needs to make sure that it has enough mana to be able to play those spells when they come up. This is why Blue/White Control for example often runs twenty-six lands.
I'm not going into any mana theories such as how many lands a mana-producing permanent counts for. That's far too in depth for this piece. Suffice it to say you want as few lands as you can get away with while still being able to play your spells in a timely manner.
Bringing it All Together Like Justin's Hand and Janet's Bustier
All of these things mana base card drawing redundancy and The Rule of Four should be taken into account when building a new deck. They help you know that you are more likely (if you do them well) or less likely (if you don't) to get a card in your deck.
It's helpful to see how good decks and good players use these things to make probabilities favor them. For example take a look at the Red/White Slide-Rift deck that Nate Heiss wrote about last week. The first thing I looked at was how many cards in the deck draw a card via cycling. If I added correctly I get twenty-eight or 46.66%. This deck is going to be drawing a lot of cards. Now how does it deal with swarms of creatures? In other words what kind of mass removal does it run? The deck has four Wrath of God two Akroma's Vengeance one Obliterate and four Starstorms. That's eleven cards out of sixty or nearly 20% that can sweep the board. In terms of effective redundancy that's well a bunch. Between the fact that nearly one in five cards can sweep the board while nearly one in two can draw a card this deck is going to find an answer to a swarm of critters pretty often.
So let's say Nate (Can I call you Nate?) is playing this deck against me in the finals of the Ohio Valley Regionals and all of a sudden my De-Scept-ive Elves deck from last week goes off. I get to cast a Hivemaster then another then a couple of other Elves. Nate's looking down the barrel of a ton of damage. But is he worried?
"'Cause he's playing you!" (Shaddup!)
While that is a good answer the better one is because he's got cycling cards galore and tons of mass removal. He cycles a Secluded Steppe. It gets him a Spark Spray. Which he cycles to get a Starstorm. Which he casts for two. I get to save one Hivemaster. I am underwhelmed.
That's not topdecking or luck or fate or kismet or karma or any other bad white enchantment. That's a good player playing a good deck. Am I suspicious of anything? Only that (1) I somehow made the finals of a regional that (b) I don't qualify to play in. The fact that he was able to find an answer to my Elves is not surprising in the least.
It's also good to understand the averages and probabilities so that you can be wary when strange things start happening. Let's go back to that One-in-Forty Spot again. Let's say that you notice a player who is consistently in a bad spot with almost no hope of winning. (See Digression the Third.) He seems to play himself into these corners where he has to rely on drawing the one card in his or her deck. This isn't a case like Nate Heiss playing that R/W Slide-Rift deck. This is a case of a player who has to get that one card in forty or lose.
When it happens the first time it could be randomness but I remember it.
When it happens a second time I get a little less comfortable.
When I see it happen for the third or fourth or fifth time in a tournament I have to figure something is up.
Why is this different from the examples above where the person drew the right card twice in a row in the One-in-Forty Spot? In those examples it was specifically noted that this was due to simple random chance. In real life we don't know that it's random. In fact the more often that an unlikely event happens the less likely it is to actually be the result of randomness. In a Magic tournament if a person is consistently topdecking to win games when any other draw would result in a loss red flags should go up everywhere.
"Wait for it . . . ."
Learn about and understand probabilities and odds and randomness. It will help make you a better deck builder. It will help make you a better player. It will also help make you less vulnerable to cheating.
As usual you've been a great audience. Now go find the lowest prime number greater than sixteen-thousand and that ends in a seven.