Me and my friend were playing a game of three players with a 52 cards deck (17, 17, 18 card each), one of the player got 4 of the same type of card (four king), I told him that he is pretty lucky and he replied that it's not that unprobable
Doing simulation with a python code I find a probability of $P \approx 0.0088$
However I cannot reproduce this result by hand, I thought first that the probabilty would be
$$P = \frac{C^{4}_{17}}{C^{17}_{52}}$$
Which is a too small number to be relevant.
Going on the base that the probability of drawing 4 same type of card by just taking 4 card out of 52 is the inverse of $C_{52}^{4}$ I felt like I needed to multiply this combinaison by an another combinaison which would represent the number of hand I can have by taking more card but I cannot put this in mathematical terms
Would someone be able to clarify this ?
The probability of getting all four $K's$ in a $17$ card hand is $$\binom {48}{13}\Big /\binom {52}{17}\approx .00879$$
What's the probability of getting any (unspecified) four of a kind? Well, you can do it exactly using Inclusion-Exclusion. We can approximate it quickly if we ignore the extremely small chance of getting $3$ or $4$ different four of a kinds. Under that approximation we get $$13\times .00879-\binom {13}2\times \binom {44}9\Big / \binom {52}{17}\approx .108$$
Which is fairly high.