I think I’ve got an answer for this challenge! Keep in mind that I haven’t played the game yet; I just downloaded the rules and print-and-play version of the cards today, so there’s a decent chance I’m wrong about something here. But, assuming that’s not the case:
The tl;dr of this is that yes, it is absolutely possible to deal 12 damage to a player on the very first turn of the game, and the chances of doing so are somewhere around 0.0048%, or once every 20,834 games.
Now, the fact that you can play Runes of Zun with up to 4 players and in teams means that there’s a few different scenarios. The simplest scenario involves just two players, though this is the also the scenario with the lowest likelihood of a turn-1-win, since there are less cards that can potentially be played in a single turn (since you can target multiple players in a turn and each player may have reactions, more players means more cards that can potentially be put into play). As such, the 1v1 scenario is the one I’ll address here. I’m also assuming that you’re playing first here; in other words, your opponent won’t have had any opportunity to play cards yet and will still be at 12 life.
With that out of the way, the answer to the first part of the challenge is actually almost trivially easy. In fact, it’s nearly solved in the given example: With an opening hand that includes 4 damage (e.g. 4 Attack, or 3 Attack + 1 Boost, or 2 Attack + 2 Boost, etc.) and one Scramble, we can hit our opponent for 4 then Scramble ourselves to draw 5 more cards. There’s 2 copies of Scramble in the deck, so all we need to do is draw a hand that’s similar to our opening hand: 4 damage and the second copy of Scramble. We hit our opponent for 4 again (8 total), then Scramble ourselves a second time, drawing 5 new cards. As long as 4 of those 5 cards deal a damage (Attacks + Boosts), we can hit our opponent for yet another 4 damage and win the game before they even get a turn.
The bonus part of this challenge (that is, the probability of this sequence occurring naturally) requires a bit more work to answer. The above scenario works only if some specific conditions are met.
- There are 2 copies of Scramble in the top 15 cards of the deck (before we deal starting hands)
- There are 0 copies of Scramble in our opponents’ hand*
- There are at least 12 damage cards in the top 20 cards of the deck (before we deal starting hands)
- There are 0 Redirects in our opponents’ hand
To find the probability of this specific turn-1-win, we just need to find the probability of each of these events occurring and multiply them together. Let me briefly explain each condition, then how we can find the probability of it being true.
Condition 1: This sequence relies on playing two copies of Scramble to refill our hands mid-turn. There are only two copies of Scramble in the entire 60-card deck, and we need both of them to be accessible to us. That means that one of them has to be in our starting hand and the other has to be somewhere in the 5 cards that are on top of the deck once both players have been deal their starting hands.
Condition 2: If our opponent has one of the 2 copies of Scramble in their hand, our sequence doesn’t work.
Condition 3: We have to deal 12 damage to our opponent, which means we need cards capable of inflicting 12 damage between our starting hand, the first 5 cards we draw off the first Scramble, and the next 5 cards we draw off the second Scramble.
Condition 4: There are 0 Redirects in our opponent’s hand. Redirect is the only reaction card capable of stopping us here, so we need to make sure our opponent doesn’t have any.
We can use what’s known as a hypergeometric distribution to find the probability of each condition occurring. A hypergeometric distribution is a function that gives you the probability of x number of successes occurring in y number of trials given the total number of successes in a population and the size of that population. For condition 1, the chances that 2 copies of Scramble will be in the top 15 cards of the deck, given that there are only 2 copies of Scramble in total and 60 cards in the entire deck is about 5.93%. We can model the chances of our opponent having 0 copies of Scramble in their starting hand given that there 2 copies in the top 15 cards of the deck (ie Condition 1 is fulfilled) by treating their starting hand as the top 5 cards of the deck; the hypergeometric distribution of 0 copies of Scramble being in the top 5 cards given that there are 2 copies total within the top 15 cards is about 42.86%. The chances of 12 damage cards (ie either Attack or Boost cards) being in the top 20 is only about 2.69%, and the chance that our opponent has no Redirects in their opening hand is about 69.94%. Multiplying all of those probabilities together gives us a total chance of about 0.0048% that we’ll deal 12 damage to our opponent in this way before they even get a turn.
Now, there are some caveats here. First, this isn’t technically the only sequence that can deal 12 damage in a turn. For example, we assumed that our opponent didn’t have any copies of Scramble, because we need both of them. The thing is, you might notice that we actually only use 14 of the 15 cards we draw in this sequence (12 damage + 2 scramble). That means we have one free slot that allows for other permutations to be feasible. It’s possible for this sequence to still work if our opponent draws one of the Scrambles as long as we draw a Steal and nab it from them. That increases our chances slightly. You may also notice that Condition 3 can be satisfied and the sequence can still fail. If there are exactly 12 damage spells in the top 20 cards of the deck and our opponent is dealt three of them, that means we’re only gonna see 9 of those damage spells; not enough to kill our opponent. However, a hand that has Attack, Boost, Share, Redirect, Scramble also deals 4 damage to our opponent (by adding a Boost to an Attack, then Sharing it with ourselves and Redirecting the Shared Attack to hit our opponent with 2 copies of the Attack+Boost), so we don’t necessarily need 12 copies of just Attacks and Boosts in the top 20. The issue with all of these different combinations is that there’s a lot of them. It’s theoretically possible for me to go through every single permutation of the 60-card deck and find each one that leads to a turn-1-win, but that would require significantly more time than I’m willing to put into this problem at the moment. That said, I so far haven’t found any turn-1-win that doesn’t require both copies of Scramble. We’ve already determined that the chances of both copies being in the top 15 cards of the deck (so that we can have one in our opening hand and then find the second copy after playing the first) are less than 6%, so I’m confident in saying that the likelihood of any particular 1v1 game of Runes of Zun resulting in the first player winning on their very first turn is between 0.0048% and 6%, though the real chance is probably much closer to that lower boundary.