It’s a numbers game!

When to Mulligan

Written by Richard Lataille

One area of uncertainty for a lot of people revolves around the starting hand – To mulligan or not to mulligan? I wanted to take a pretty straightforward example of a Dark Phoenix ramp deck and discuss when to mulligan and when to keep your hand. In this example, we’re going to use a standard DP ramp deck not using Pixie and Singularity. Our key cards will be 4 Academy and 4 Madripoor (let’s call them “yellow equivalents” or “YEs”). You need some combination of 2 of these 8 cards to perform your first ramp which will really kick off your game. I’ve seen posts and discussions where people advocate for either keeping when you have at least 2 of these cards in your starting hand, or keeping if you have at least 1. So this is our test – should you mulligan with 1 yellow equivalent, or 2? So first let’s get our basic numbers: 60 cards in a deck 7 cards in the starting hand 8 yellow equivalents in the deck 2 yellow equivalents required to ramp Here are our chances for each quantity of yellow equivalents in the initial draw: 0 – 35% 1 – 42% 2 – 18.5% 3 or more – 4.5% We have one choice to make, Mulligan or Keep. Strategy One – Keep with at least 1 Yellow Equivalent Based on our initial odds, our chance of getting no yellow equivalents, and thus choosing mulligan, is 35%. Our chance of getting at least 1 yellow equivalent, and thus keeping, is 65% (42% + 18.5% + 4.5%). When we mulligan, our odds of getting each quantity of yellow equivalents is the same as out initial draw; however, to calculate our overall odds using this strategy, we need to multiply our initial odds by the chance of choosing mulligan.

We end up with the following possibilities when using this strategy:

Draw 1 YE in initial hand and choose to keep – 42% Draw 2 YE in initial hand and choose to keep – 18.5%

Draw 3 or more YE in initial hand and choose keep – 4.5%

Mulligan, and then draw 0 YE again – 12.25% (35% * 35%)

Mulligan, and then draw 1 YE – 14.75% (35% * 42%) Mulligan, and then draw 2 YE – 6.5% (35% * 18.5%)

Mulligan, and then draw 3 or more YE – 1.5% (35% * 4.5%)

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When we consolidate our odds using this strategy, our overall odds are as follows: 0 – 12.25% 1 – 56.75% 2 – 25% 3 or more – 6% This gives us a 31% chance of ramping on turn one and a fairly low chance at having no yellow equivalents to start. Strategy Two – Keep with at least 2 Yellow Equivalent With this strategy, our chance of choosing mulligan, is 77% (35% chance of 0 + 42% of 1). Our chance of getting at least 2 yellow equivalents, and thus keeping, is 23% (18.5% chance of 2 + 4.5% chance of 3 or more). Again, when we mulligan, our odds of getting each quantity of yellow equivalents is the same as out initial draw; however, to calculate our overall odds using this strategy, we need to multiply our initial odds by the chance of choosing mulligan (again, 77%).

We end up with the following possibilities when using this strategy:

Draw 2 YE in initial hand and choose to keep – 18.5%

Draw 3 or more YE in initial hand and choose keep – 4.5%

Mulligan, and then draw 0 YE again – 27% (77% * 35%) Mulligan, and then draw 1 YE – 32.25% (77% * 42%)

Mulligan, and then draw 2 YE – 14.25% (77% * 18.5%)

Mulligan, and then draw 3 or more YE – 3.5% (77% * 4.5%)

When we add our chances up using this strategy, our overall odds are as follows:

0 – 27%
1 –32.25%
2 – 32.75%
3 or more – 8%

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With this strategy, we end up with a solid 40% chance of ramping on turn one, which is approximately 10% more than the first strategy. On the other hand, we also have a 27% chance of starting the game with no yellow equivalents at all.

So, knowing what you know now, what do you think? Which is the superior strategy?

Please feel free to provide feedback and critique my math. I am by no means a math whiz; I simply find the statistics behind the game very interesting. I have no problem with people leverage my numbers; however, please provide a reference to me and/or this article.math-is-46ecpl

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