Starting Hand Odds: How Many Two-Drops is Enough?
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Knowing the right amount of low-cost cards needed to consistently secure a viable starting hand is crucial; without the little guys, a control deck falls apart and an aggressive deck never gets going. Luckily, the following tables can help you build your decks accordingly:
# = the number of desired starting-hand cards that you put in your deck
1st = the odds of securing a particular number of desired cards when going first
2nd = the odds of securing a particular number of desired cards when going second
Avg = the average of “1st” and “2nd
The first two tables are likely the most useful, as it’s not particularly necessary in most decks to have low-cost cards available on turn 3 and/or 4. However, it’s interesting to look at the Warlock deck played in the most recent King of the Hill. With eight 1-cost minions, there’s a 93.7% chance of securing one of them in your starting hand, and a 65% chance of getting two. Similarly, with twelve combined 1- and 2-cost minions, there’s a 99% chance of getting at least one in your starting hand, and an 88% chance of getting two or more. There’s a reason why it’s one of the best decks in the world, and the math bears that out.
*denotes values that aren’t perfectly 100 or 0, E.G. 99.999999813
Note: the tables assume that you mulligan as aggressively as possible; I.E., you mulligan any card that isn’t on your “desired” list
The odds of getting at least one “desired” card in your starting hand (including the card acquired after the mulligan):
[su_tabs][su_tab title=”At Least 1 Card”]
| # | 1st | 2nd | Avg |
| 1 | 23% | 29.5% | 26.2% |
| 2 | 41.3% | 51% | 46.1% |
| 3 | 55.7% | 66.4% | 61.1% |
| 4 | 67% | 77.4% | 72.2% |
| 5 | 75.7% | 85.1% | 80.4% |
| 6 | 82.4% | 90.3% | 86.4% |
| 7 | 87.4% | 93.9% | 90.6% |
| 8 | 91.2% | 96.2% | 93.7% |
| 9 | 93.9% | 97.7% | 95.8% |
| 10 | 95.9% | 98.7% | 97.3% |
| 11 | 97.3% | 99.3% | 98.3% |
| 12 | 98.3% | 99.6% | 98.9% |
| 13 | 98.9% | 99.8% | 99.4% |
| 14 | 99.4% | 99.9% | 99.6% |
| 15 | 99.6% | 100%* | 99.8% |
| 16 | 99.8% | 100%* | 99.9% |
| 17 | 99.9% | 100%* | 99.9% |
| 18 | 99.9% | 100%* | 100%* |
| 19 | 100%* | 100%* | 100%* |
| 20 | 100%* | 100%* | 100%* |
| 21 | 100%* | 100%* | 100%* |
| 22 | 100%* | 100%* | 100%* |
| 23 | 100%* | 100% | 100%* |
| 24 | 100%* | 100% | 100%* |
| 25 | 100% | 100% | 100% |
| 26 | 100% | 100% | 100% |
| 27 | 100% | 100% | 100% |
| 28 | 100% | 100% | 100% |
| 29 | 100% | 100% | 100% |
| 30 | 100% | 100% | 100% |
[/su_tab] [su_tab title=”At Least 2 Cards”]
| # | 1st | 2nd | Avg |
| 1 | 0% | 0% | 0% |
| 2 | 4% | 7.1% | 5.6% |
| 3 | 10.7% | 18.1% | 14.4% |
| 4 | 19.2% | 30.6% | 24.9% |
| 5 | 28.4% | 43.1% | 35.8% |
| 6 | 38% | 54.8% | 46.4% |
| 7 | 47.3% | 65.1% | 56.2% |
| 8 | 56% | 73.8% | 64.9% |
| 9 | 64% | 80.8% | 72.4% |
| 10 | 71.2% | 86.4% | 78.8% |
| 11 | 77.3% | 90.6% | 84% |
| 12 | 82.6% | 93.8% | 88.2% |
| 13 | 86.9% | 96% | 91.5% |
| 14 | 90.4% | 97.5% | 94% |
| 15 | 93.2% | 98.5% | 95.9% |
| 16 | 95.3% | 99.2% | 97.2% |
| 17 | 96.9% | 99.6% | 98.2% |
| 18 | 98% | 99.8% | 98.9% |
| 19 | 98.8% | 99.9% | 99.4% |
| 20 | 99.3% | 100%* | 99.6% |
| 21 | 99.6% | 100%* | 99.8% |
| 22 | 99.8% | 100%* | 99.9% |
| 23 | 99.9% | 100%* | 100%* |
| 24 | 100%* | 100%* | 100%* |
| 25 | 100%* | 100% | 100%* |
| 26 | 100%* | 100% | 100%* |
| 27 | 100% | 100% | 100% |
| 28 | 100% | 100% | 100% |
| 29 | 100% | 100% | 100% |
| 30 | 100% | 100% | 100% |
[/su_tab] [su_tab title=”At Least 3 Cards”]
| # | 1st | 2nd | Avg |
| 1 | 0% | 0% | 0% |
| 2 | 0% | 0% | 0% |
| 3 | 0.4% | 1.3% | 0.9% |
| 4 | 1.5% | 4.6% | 3.1% |
| 5 | 3.4% | 9.9% | 6.7% |
| 6 | 6.3% | 16.9% | 11.6% |
| 7 | 10.1% | 25.2% | 17.7% |
| 8 | 14.8% | 34.3% | 24.6% |
| 9 | 20.3% | 43.7% | 32% |
| 10 | 26.4% | 53% | 39.7% |
| 11 | 33% | 61.7% | 47.4% |
| 12 | 40% | 69.6% | 54.8% |
| 13 | 47% | 76.6% | 61.8% |
| 14 | 54.1% | 82.5% | 68.3% |
| 15 | 60.9% | 87.3% | 74.1% |
| 16 | 67.4% | 91.1% | 79.3% |
| 17 | 73.5% | 94.1% | 83.8% |
| 18 | 78.9% | 96.2% | 87.6% |
| 19 | 83.8% | 97.7% | 90.7% |
| 20 | 87.9% | 98.7% | 93.3% |
| 21 | 91.4% | 99.3% | 95.4% |
| 22 | 94.2% | 99.7% | 96.9% |
| 23 | 96.3% | 99.9% | 98.1% |
| 24 | 97.8% | 100%* | 98.9% |
| 25 | 98.9% | 100%* | 99.4% |
| 26 | 99.5% | 100%* | 99.8% |
| 27 | 99.8% | 100% | 99.9% |
| 28 | 100%* | 100% | 100%* |
| 29 | 100% | 100% | 100% |
| 30 | 100% | 100% | 100% |
[/su_tab] [su_tab title=”At Least 4 Cards”]
| # | 1st | 2nd | Avg |
| 1 | 0% | 0% | 0% |
| 2 | 0% | 0% | 0% |
| 3 | 0% | 0% | 0% |
| 4 | 0%* | 0.1% | 0.1% |
| 5 | 0.1% | 0.6% | 0.4% |
| 6 | 0.4% | 1.7% | 1% |
| 7 | 0.8% | 3.5% | 2.1% |
| 8 | 1.5% | 6.1% | 3.8% |
| 9 | 2.6% | 9.7% | 6.1% |
| 10 | 4% | 14.2% | 9.1% |
| 11 | 5.9% | 19.6% | 12.7% |
| 12 | 8.3% | 25.7% | 17% |
| 13 | 11.2% | 32.3% | 21.8% |
| 14 | 14.6% | 39.4% | 27% |
| 15 | 18.5% | 46.7% | 32.6% |
| 16 | 23% | 54% | 38.5% |
| 17 | 27.9% | 61.1% | 44.5% |
| 18 | 33.2% | 67.9% | 50.5% |
| 19 | 38.9% | 74.2% | 56.5% |
| 20 | 44.8% | 79.9% | 62.4% |
| 21 | 51% | 84.9% | 68% |
| 22 | 57.3% | 89.2% | 73.3% |
| 23 | 63.6% | 92.7% | 78.2% |
| 24 | 69.9% | 95.5% | 82.7% |
| 25 | 75.9% | 97.5% | 86.7% |
| 26 | 81.7% | 98.9% | 90.3% |
| 27 | 87% | 99.7% | 93.4% |
| 28 | 92% | 100%* | 96% |
| 29 | 96.3% | 100% | 98.1% |
| 30 | 100% | 100% | 100% |
[/su_tab][/su_tabs]
SIMILAR ARTICLES
I don’t know if I’m right, but I’m almost sure maths are not ok. Let me explain; If I understood what you say “If you put 20 desired cards in your deck you get 100% chance to get at least one at the start”. To do the maths i will assume I’m going second, so your starting hand is 4+1.
The odds to get 0 desired cards would be:
(i) First 4 draw -> 10/30 * 9/30 * 8/30 * 7/30. Now I suppose you discard all the cards and you can’t draw the same cards again.
(ii) Second 4 draw -> 6/30 * 5/30 * 4/30 * 3/30
(iii) Turn 1 draw-> 6/30
So, the odds of getting an “awful” hand with 20 desired cards in your deck are:
(i) 0.504 * (ii) 0.036 * (iii) 0.6 = 0.0108864
The average is 1 “awful” hand every 100 games.
PS: Sorry for my english.
[Edit] So, the odds of getting an “awful” hand with 20 desired cards in your deck are:
(i) 0.00622222222* (ii) 0.00044444444 * (iii) 0.2= 5.53086414e-7
The average is 1 “awful” hand every 20 000 000 games.
PS: It’s almost 100% but it is not.
I think you just answered your own question =)
1 minus “5.53086414e-7″ is not 1.000, sure, but it’s not .999, either. I made these tables using Microsoft Excel, and I formatted the tables to say, “display as a percentage, and show one decimal place.” The program thereby automatically rounded to the nearest thousandth.
When I submitted the article, I wasn’t sure how Sympatico would edit the tables (speaking of which, thanks for making my articles look so pretty, Sympatico!). If he placed the tables side-by-side like in my other article (https://ihearthu.com/mulligan-math-what-are-the-odds/), then there would have been no room to write “99.99995%” And even in this format, with the extra room, it’s arguably better without “99.999999813%” and “.000000001%” littering the tables; at some point, you have to make it easy to read/scan over. I think most people are okay with a 1-in-20 million chance being labeled as “0” and vice versa.
It’s always nice to see people doing maths out there, and i don’t want to disregard your job (wich is pretty good), but i wanted to specify that was not a 100% and we all know the Murphy’s law.
Thanks for the kind words. You can understand why I’d get defensive when a comment comes that says “I’m almost sure [you’re wrong].” There was a similar comment from the reddit thread that Sympatico started, and it was clear the commenter had just neglected to read carefully enough.
I agree, but that specification should have occurred with an asterisk. Solid job nonetheless.
Ok, I had Sympatico put the asterisks in.
The tabs are broken. The code is appearing instead of the visual effect.