On this fourth article of my Probability in MTG series, I discuss what it takes to be able to “go infinite” in MTG: Arena BO1 drafts.

What are MTG: Arena BO1 drafts?

Arena drafts are events that cost 5000 gold or 750 gems. You get to play best-of-one (BO1) matches until your either accumulate 7 wins or 3 losses and gain prizes depending on your number of wins.

What is “going infinite?”

In the current MTG:Arena BO1 draft event, winning 6 times or more ensures that you gain enough prizes to join another event. Winning fewer times than that allows you to recoup some of your entrance fee, depending on how many games you won. Going infinite means being able to win enough games on average such that on average, you gain more gems than you lose. This is described as having positive expected value (EV). I wrote a previous article on EV here but the examples used are more related to MTGO than Arena (the concept of EV is the same in either platform).

The Math: Negative Binomial Distribution

th loss. In our situation, we want to be able to determine the probability of winning X number of times before our 3rd loss. This probability is affected only by our probability of winning; a number which we have to assume. With this distribution defined, we can then use it to compute our expected value for playing the game. The following table provides a summary of expected values across different win rates. I discussed probability distributions and how they relate to MTG in another previous article . Here I will just cut to the chase and say that in the MTG:Arena BO1 draft event, the probability distribution of interest is the negative binomial . This distribution fits the random variable that counts that number of successes before the rloss. In our situation, we want to be able to determine the probability of winning X number of times before our 3loss. This probability is affected only by our probability of winning; a number which we have to assume. With this distribution defined, we can then use it to compute our expected value for playing the game. The following table provides a summary of expected values across different win rates.

Win Rate 30% 40% 50% 60% 75% 80% EV (in Gems) -597 -518 -403 -251 5 80

As you can see from the table, a win rate of about 75% is needed in order to “go infinite.” That is, if you believe that you win at least 75% of your games in draft, then by playing a large number of games, you should end up gaining more gems than you lose. A caveat of this computation is that expected value is an average. That is, there is considerable variance involved in that estimate so that even if you do have a 75% win percentage, you will still need to bankroll your drafts at the beginning in order for you to be able to weather out early losses.

So what does all of this mean?

First, going infinite on BO1 drafts is no easy feat. However, it is also far from impossible. If you are just starting out in MTG, playing more drafts should improve your understanding of the format and push your win rate up. A nice element of draft is that your win rate is not affected by your wallet. That is, you do not need to have a tier 1 constructed deck to win consistently.

Second, even if you’re not yet quite at 75%, doing drafts is still a good idea. An EV of -251 gems means that if you start on 9,200 gems ($49.99), you’d be able to do about 36 drafts on average before your stash of gems gets depleted. That means you’d have opened about 108 boosters, not counting the ones you win from prizes. This is valued at 18,000 gems which is double of the number of packs that 9200 gems would have been able to buy. Even if your win rate is just 30%, you’d still end up with the same value on average as the number of packs that 9200 gems would have been able to buy. One factor that I did not take into account here is that of wildcards, which you would get more of by just opening packs than by drafting. However, the difference in raw value should be able to make up for this as long as your win rate is not too low.

Finally, drafting looks like a solid way to build one’s collection in preparation for standard. Daily rewards enable one to accumulate enough gold to do a draft a week. In my opinion, similar to how MTGO has a golden rule to never to open packs, it is in the best interest for Arena players to never use gold/gems for buying packs. Perhaps with the exception of really needing those wildcards to make a specific deck quickly.

Experience disclosure

I have only done just 4 GRN drafts on Arena so far, myself (I still mainly play MTGO). My records were 7-1, 3-3, 2-3, 7-2. So This means that from an initial input of 1550 gems, I am at 950 gems (losing 600 gems after 4 drafts so far) with enough gems to buy into another draft without needing to purchase more gems yet. That’s with a sample win rate of about 68%.