Magic Math – Krark’s Thumb and Mana Clash in Legacy

Do you ever feel lucky? Do you ever wish that Magic relied more on chance elements? Do you feel that the more unlikely the victory, the more memorable the success? Well, then I have the perfect deck for you!

Legacy Coin Flip

Ben Bryner

Ben Bryer was brave enough to register this deck for Grand Prix Seattle. Although he only went 2-6, the coverage team still found his deck interesting enough to do a Deck Tech video on his creation. I can’t blame them.

Ben’s build may not look perfect—Scalding Tarn and Volcanic Island would be better than Mana Confluence, and the 4 Plains in the sideboard don’take much sense to me either—but these are likely budget card choices. And more importantly, the deck looks like it would be a blast to play.

Krark’s Thumb

Krark’s Thumb are the heart of the deck. Without Krark’s Thumb, the probability of flipping heads or tails is obviously 50%. With Krark’s Thumb, you’ll flip 2 coins and choose one of the two outcomes. There are 4 possible outcomes in total, all of which occur with the same probability:

4. Tails + Tails

So if you prefer to see heads, then you will be able to choose heads in 75% of the cases. This substantially increases the likelihood of getting a good outcome: Stitch in Time turns into a Time Walk 75% of the time, Puppet’s Verdict will give you the desired effect 75% of the time, and Planar Chaos will allow your spell to resolve in 75% of the cases. Sure, it’s 100% with Vexing Shusher, but that doesn’t have the same excitement value.

Let’s run the numbers on what Krark’s Thumb can do for the rest of the deck.

Fiery Gambit

Without Krark’s Thumb in play, the probability of winning one flip is 50%, the probability of winning two flips is 25%, and the probability of winning three flips is 12.5%. So if you keep flipping, then you’ll be able to draw 1.13 cards in expectation.

With Krark’s Thumb in play, however, the probability of winning one flip is 75%, the probability of winning two flips is 56.25%, and the probability of winning three flips is 42.19%. As a result, you’ll be able to draw 3.80 cards in expectation if you never choose to stop flipping. Much better.

Game of Chaos

Since the stakes are doubled each time (it goes 1-2-4-8-16-32) you just need to win 6 coin flips in a row to win the game right away if your opponent starts at 20. Assuming that your opponent always chooses to stop the process if possible, the probability of an immediate win can be determined in a similar way as for Fiery Gambit.

Taking the sixth power of 0.5 or 0.75 reveals that the probability of an immediate win is 1.56% when you don’t have Krark’s Thumb in play and 17.80% when you do. The artifact makes a huge difference!

Mana Clash

Now we get to the real fun. When you have Krark’s Thumb in play, you will flip your two coins at the same time as your opponent will flip theirs, and you get to see the result of all flips before you decide which of your two coins to ignore. This gives you a reasonable amount of control over the outcomes.

(It appears that this is not implemented correctly on Magic Online, by the way.)

The question I will tackle is as follows: How likely are you to win immediately with a Thumb-powered Mana Clash? Moreover, how likely is an immediate loss or draw? (Since state-based effects are not checked during resolution of Mana Clash, one of the two players is allowed to fall down to 0 life while the coin flipping continues. It is possible that both players are at a negative life total when both coins come up heads, in which case the game is a draw.)

You can think of the situation as a discrete-time, absorbing Markov chain. To me, that implies some nice, recreational math. The solutions and explanations are below.

These formulas can be easily implemented in four 21 x 21 spreadsheets to get numerical outcomes. The below table provides the percentage probability that the game will end in a win for you, depending on what the life totals of the two players are when Mana Clash is cast.

If you want the numerical outcomes for drawing, losing, or continuing, then you can find those in this Google spreadsheet.

So all in all, if both players start at 20 life, then there is a 0.77% probability that you win, a 0.52% probability that you lose, a 0.38% probability that the game ends in a draw, and a 98.33% probability that there will be 2 heads before anyone falls to zero life. Not the most reliable of potential turn-1 kills (especially if you consider that you need to draw 2 Lotus Petals to even cast Krark’s Thumb and Mana Clash on turn 1), but it’s not completely unthinkable.

If some damage had early been done beforehand (with Legacy staple Karplusan Minotaur, perhaps) then the numbers change. If, say, your opponent is at 3 life and you are at 9, then your thumb-powered Mana Clash will win the game with probability 50.13%, lose with probability 0.10%, draw with probability 1.07%, and continue the game with both players alive with probability 48.70%.

Still not exactly a mark of consistency, so I don’t see this deck winning major tournaments anytime soon—but it’s a fine choice if you feel you need some more randomness in your life. Perhaps I’ll test my luck at an upcoming Legacy event…