# Chances of getting a card in Commander

I have four different tutors in my Commander deck of 100 cards. What are the odds that I will draw at least one of them in my first seven cards drawn for my hand? Back in the day I knew how to do this, but haven't touched statistics in many years.

The easy method is to multiply the probability that each of your starting 7 isn't one of the cards you want, then subtract that total from 1.

Assuming you desire any of the tutors, or the card to be tutored, there are 94 cards that aren't what you want out of the 99 in your deck when you start drawing.

1 - ((94/99) * (93/98) * (92/97) * (91/96) * (90/95) * (89/94) * (88/93))

= 31%

• I think that including "the card to be tutored" here is not valid. The value of a tutor in many deck is that it can be used reactively to get whatever card fits the situation best. So in many cases, there isn't a specific "card to be tutored" at all at the start of the game. Aug 16 at 16:22
• I think you are right bout the "card to be tutored", as I don't think I know on turn one what I need. But an approximation is all I was looking for. So 31% is a good thing to know. Thanks.
– bobW
Aug 16 at 18:21
• You can change the formula to count only the 4 tutors just by increasing the numerator of each fraction by 1. In that case, it works out to about 26%. Aug 16 at 18:42

Questions such as this can be answered with a hypergeometric calculator, since the underlying distribution in mathematics is the hypergeometric distribution.

MtG hypergeometric calculator

1. Population size = 99 (since there are only 99 cards in a Commander deck, the last card is the Commander in the Command zone)
2. Sample size = 7 (your opening hand size)
3. Successes in population = 4 (you want to draw the four tutors; if you also include the card to be tutored for then successes in population would be 5)
4. Successes in Sample = 1 (you only need one of them in your opening hand)

Chance to draw 1 or more of the wanted card 25.8%

Chance to draw exactly 1 of the wanted card 23.4%

Chance to draw 1 or less of the wanted card 97.6%

Chance to draw 0 of the wanted card 74.2%

• Note that the first (chance to draw 1 or more) and the last (chance to draw 0) sum to 100% - because ultimately, these are the only two outcomes - either you got one or you didn't. Aug 17 at 1:18
• You can also use the fact that 100 is a fairly large number to do some rough math in your head - 7 cards, 4 successes, (4*7/100) = 28%. Obviously the exact math is much more precise, but that will get you a general ballpark. Aug 17 at 1:19
• These numbers are a little off because a commander deck's library starts with 99 cards, not 100. The 100th card is the commander, and it is not in the library. Aug 17 at 1:19
• @murgatroid99 good point, amending population size to 99. Aug 17 at 1:21