Ok, we have next data:
Chance to level up - 15%
Number of tries - 6
Bernoulli theorem to the rescue!
1-(1-0.15)^6 = 1 - 0.85^6 = 1 - 0.377 = 0.623 = 62% that you level up skill after using 6 tries.
But here is the trick - we can raise level twice in this manner.
Formula here -

Chance of raising level twice in six tries = P(2|6)=C(2|6)*0.15^2*0.85^4=6*0.0225*0.522=0.07 = 7%
So chance of getting one or two level ups is 62+7 = 69%
You can calculate chances of getting three level ups and four and add them yourself.
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Summary: Use 6 cards to raise level of skill. It's safer and use less math)
Midas aka Howrus. GM of Fairy Tails.