Destiny has a pretty friendly mulligan rule. But just how often will you get the cards you are looking for in your opening hand?

Say you are playing a deck with Jango Fett and you always want a Jetpack in your opening hand.

Assuming you are playing 2 copies of Jetpack, your odds of getting at least one in your first draw is 31% .

If you mulligan exclusively for a Jetpack, your odds of getting a Jetpack after the mulligan is 52% .

So on average, in half of your games you will get at least one Jetpack in your opening hand after the mulligan.

If you are looking for either one of two different cards, say Jetpack or Holdout Blaster.

Odds of getting at least one in your first draw: 54%

Odds after an exclusive mulligan of getting at least one: 79%

So on average, 4 out of every 5 games you will get at least one Jetpack or Holdout Blaster in your opening hand after the mulligan.

If you are looking for either one of three different cards, say Jetpack, Holdout Blaster, F-11D Blaster.

Odds of getting at least one in your first draw: 70%

Odds after an exclusive mulligan of getting at least one: 90%

So on average, 9 out of every 10 games you will get at least one of those three cards in your opening hand after the mulligan.

Conclusion: Obviously the more card you have in your deck that you want in your opening hand, the more likely you are to get at least one. But the exact numbers are good to know as well. So build your decks and mulligan accordingly. If your deck hinges on a first turn Sith Holocron, you won't get it about half of the time.

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*PS: I hope my math is correct. Here it is if anyone wants to check.

(28 choose 5) / (30 choose 5) is the odds of not getting at least one copy of a card that you have two of in your deck when drawing 4 cards.

1- (28 choose 5) / (30 choose 5) = .31

1- ((28 choose 5) / (30 choose 5)) ² = .52

1- (26 choose 5) / (30 choose 5) = .54

1- ((26 choose 5) / (30 choose 5)) ² = .79

1- (24 choose 5) / (30 choose 5) = .70

1- ((24 choose 5) / (30 choose 5)) ² = .90

Yes, I believe that your math is correct. 31% chance of pulling one jetpack on the first draw and 52% on the second draw given they are independent events if you discard all 5 cards.

Couldn't begin to check your math, but this is very interesting to know. I know it's not something to bank on in games featuring 60+ card decks, but I like the idea of building in several potential "kickoff" cards...

Will re-examine my decks presently.

I'm not sure what all the math is written out when you say (28 choose 5).

The way I would calculate it is start the way you did, determine the odds you don't get the card you want. But I thought about calculating it just drawing one card at a time, for a total of five times, without replacement for the pre-mulligan part. So the first card is 28/30 chance it won't be one I want, but I don't put that card back in the deck before my second draw so now my second card is 27/29 chance that it isn't what I want. The third card is now 26/28. The fourth card is 25/27. The final card is now 24/26. That had to be true for the mulligan to be necessary, so I know I multiply this by the mulligan percentage I'll end up calculating. And since I mulligan all 5, it ends up being identical. Then I subtracted the chance of not what I wanted to have happen from one.

Odds of getting either copy of one specific card after mulliganning every card that isn't that one card: 1 - (28/30*27/29*26/28*25/27*24/26)^2 = 0.52437574317.

Given that one matched up, I trust your others probably do as well.

I can confirm that the math is correct.

Let's say that your deck has 2 Sith Holocrons and 2 each of Mind Probe, Force Throw and Immobilize. What are the odds that you'll draw at least one Holocron AND at least two of your six blue abilities?

We'll say that there are a total of 2 possible ways to choose 1 holocron. You either pick one or the other, right? Mathematically, that looks like "2 choose 1", which equals 2. We'll further say that there are a total of 15 possible ways to choose two blue abilities. Mathematically, that looks like "6 choose 2". We've now covered the first 3 cards. We still need to figure out how many ways we can draw the remaining 2. We have 27 cards remaining in the deck, so we'll do "27 choose 2", which equals 351. There are 351 different 2 card combinations you can draw when you have 27 cards to pick from.

So... to add this all together...

(2 ways to draw 1 of your 2 Holocrons) * (15 ways to draw 2 of your 6 blue abilities) * (351 ways to draw an extra 2 of your 27 remaining cards) = 10,530 possible card drawing combinations where you drew at least 1 Holocron and at least 2 abilities.

We now know that we have 10,530 possible initial hands with what we want. We also know that there are 142,506 total possible initial hands (30 choose 5). The odds of pulling at least one Holocron AND at least 2 of 6 blue abilities is 7.39% (10,530/142,506).

Assuming 6 blue abilities in your deck...

The odds of drawing at least 1 Holocron AND at least...

• 1 blue ability is... 27.59%
• 2 blue abilities is... 7.39%
• 3 blue abilities is... 0.73%
• 4 blue abilities is... 0.02%

Assuming 8 blue abilities in your deck...

The odds of drawing at least 1 Holocron AND at least...

• 1 blue ability is... 36.78%
• 2 blue abilities is... 13.79%
• 3 blue abilities is... 2.04%
• 4 blue abilities is... 0.10%

Of course, none of this takes into consideration the mulligan mechanic. The mulligan mechanic with multiple cards you're looking for gets a little complicated.

We really have three scenarios to look at...

1. Didn't get either a Holocron or an ability, drop our hand fully and redraw.
2. Got a Holocron, but not an ability. Drop 4 cards and redraw.
3. Got an ability, but not a Holocron. Drop 4 cards and redraw.

I don't have the full time to do the math right now, but it can get pretty complex depending on what you draw and what you decide to discard. I'll probably type up an actual full analysis and put it up later for those of you who are interested.

If you google "30 choose 5" it will give you 142506, and that is the number of possible different 5 card hands from a 30 card deck. So "28 choose 5" is the number of different 5 card hands that do not contain a Jetpack (in my example), which is 98280. So 98280/142506 gives you the percentage of 5 card hands that do not contain a Jetpack, which is 0.69. And 1-0.69 = .31, the percentage of 5 card hands that contain at least one Jetpack.

Just played in a tournament 16 people.

After 3 rounds me and the other 3-0 player are playing, both with Jango. I run 8 upgrades he was running 10 or 12. We both Mulligan 5 cards looking for any uograde (let alone jet pack) turn one neither of us have an upgrade... I use all 5 of my cards draw 5 new next turn.... Still nothing, he gets a jetpack.

So my first 15 cards no upgrades, his first 10 no upgrades...

I hope my math was right. Lol... I'm super tired.

Just played in a tournament 16 people.

After 3 rounds me and the other 3-0 player are playing, both with Jango. I run 8 upgrades he was running 10 or 12. We both Mulligan 5 cards looking for any uograde (let alone jet pack) turn one neither of us have an upgrade... I use all 5 of my cards draw 5 new next turn.... Still nothing, he gets a jetpack.

So my first 15 cards no upgrades, his first 10 no upgrades...

Well, there's the trouble with math. It's all probabilities.

Which can be summed up pretty quickly: it's probably going to screw you.

Just played in a tournament 16 people.

After 3 rounds me and the other 3-0 player are playing, both with Jango. I run 8 upgrades he was running 10 or 12. We both Mulligan 5 cards looking for any uograde (let alone jet pack) turn one neither of us have an upgrade... I use all 5 of my cards draw 5 new next turn.... Still nothing, he gets a jetpack.

So my first 15 cards no upgrades, his first 10 no upgrades...

Well, there's the trouble with math. It's all probabilities.

Which can be summed up pretty quickly: it's probably going to screw you.

At least in this case it screwed both of us for the most part.