Inspired by a game of solo Spades I recently lost. In a four player game where all 52 cards are distributed evenly. If the lowest card I have in a suit is the 5, what is the probability the 2, 3, and 4 are distributed evenly among the other three players?
I only had a few seconds to bet, but my intuition would be that the odds were pretty low. If the cards were evenly distributed, I would almost definitely lose (unless another player did something really dumb). If not, I was guaranteed a win (unless I did something stupid). I bet on my intuition but ended up losing.
Was my intuition correct and I just go unlucky or was I wrong?
Total number of ways of distributing the $3$ cards to your opponents:
$$3^3$$.
Favorable outcomes (even distribution):
$$3!$$
Therefore required Probability is:
$$\frac{6}{27}$$ $$=\frac{2}{9}$$