you have 27 cards, labeled 1-27, face down in a deck and shuffled. you pick 4 completely random cards

[daveh7777]

To begin, I am not currently in high school or college, this problem is one that was posed to me by a friend of mine trying to optimally play a video game where there is a bank of 27 skills, and each time you level up you get one choice out of 4 random skills, for 8 levels. That friend wants to know the chance of getting any specific skill in a run.
I will word the problem as follows:

you have 27 cards, labeled 1-27, face down in a deck and shuffled.
you pick 4 completely random cards from the deck, and look at them. if one of the cards is #1, keep it, and put the other 3 back in the deck and shuffle it.
if one of those cards is NOT #1, pick a random card from the 4 you drew, and put the remaining 3 back in the deck and shuffle it.
repeat this process 7 more times, until you have 8 cards in your hand and 19 in the deck.
what is the probability that one of the 8 cards is the #1 card?

to be clear: one of the 4 cards in your first pick could be the #1 card, making the remaining 7 draws just random cards.
there is also a chance that you will not get the #1 card at all.

 

[daveh7777]

IMPORTANT:

There is a lot I already know. on the first draw, you simply have a 4/27 chance of getting the #1 card. if you don't get it, you then have a 4/26 chance the next hand, and so on. for each of the 8 draws, your individual probabilities are:

draw 1: 14.8%
draw 2: 15.4%
draw 3: 16.0%
draw 4: 16.7%
draw 5: 17.4%
draw 6: 18.2%
draw 7: 19.1%
draw 8; 20.%

so you have a 14.8%, if you don't get it you then have a 15.4%, and so on

also, I coded a program to run this scenario over and over, and out of 1000 games, the #1 card was picked roughly 800 times or 80%, however I still can not mathematically reach that number, so I would really like an equation of some kind to find an exact probability.