It's probably less than you think.
http://stattrek.com/online-calculator/hypergeometric.aspx
Population size: 30 card deck
Number of successes: 2 of each card
Sample size: 5 in your starting hand
Number of successes in sample: 1
The odds of drawing one of any card you have 2 of in your deck are about 28.7% so less than 1/3 of the time. Keep any of those cards and try to draw again and it goes down by about 4-5% for each card you keep.
This isn't correct. I think that looking for exactly 1 success is skewing the sample, because it's ignoring cases where you draw both copies.
If you're looking for a single copy of a card, you've got a 31% chance of getting it in your opening hand. If you mulligan aggressively for that card, it's 52.4% or so.
Even combos aren't too bad - if you're looking for a single copy of two cards you've got about a 24% chance of hitting it with an aggressive mulligan.
From interviews with the designer, they've very much intended that draws are more reliable than a lot of CCGs. Between the flexible mulligan and the ability to potentially cycle your hand every turn, the odds are VERY high that you're going to get the cards you need, and quickly. Hopefully they'll remember that while they're testing, but good bad or otherwise it is an essential part of the game design.
True, I did grab the wrong stat, because there's about a 3% chance you draw both copies and you'd want AT LEAST one of a specific card so you have a 31% to draw at least one copy that you have 2 of in your deck.
Not sure what you mean by mulligan "aggressively", we only get one mulligan, and If you put all 5 cards back and draw again, you still have a 31% chance to draw at least one of those cards.
I haven't done the math but I'd think that getting at least 1 copy of 2 different cards would be lower than 24%, you may be right though.