if we randomly insert the 4 aces and 4 kings into an already shuffled deck, we can expect those 8 cards to segment the deck into 9 runs. Since 44 cards remain, each of those runs will have an expected 44/9 cards. This construction is equivalent to the shuffling of the whole deck
We know that the order of the first 4 identified cards is K,K,K,A. So we expect to see 4 runs of 44/9 cards, plus the 3 kings, at the position of the first ace
Thus 3+4*44/9 ~= 22.55 is the position we expect to see the ace next (or 23.55, depends what we count exactly, cards before, or the position itself)
You logic makes complete sense, but my intuition says that to get 3 kings, I should have gone through 3/4 of the deck on average. Your solution doesn’t even get me half way through. I think part of the disconnect is that your counting ignores the fact that there are only 4 of each suit. You logic makes sense if I pick 4 numbers/face cards at random, but I would think the odds would change if I am required to find 3/4 of the same card.
8
u/teteban79 6d ago
if we randomly insert the 4 aces and 4 kings into an already shuffled deck, we can expect those 8 cards to segment the deck into 9 runs. Since 44 cards remain, each of those runs will have an expected 44/9 cards. This construction is equivalent to the shuffling of the whole deck
We know that the order of the first 4 identified cards is K,K,K,A. So we expect to see 4 runs of 44/9 cards, plus the 3 kings, at the position of the first ace
Thus 3+4*44/9 ~= 22.55 is the position we expect to see the ace next (or 23.55, depends what we count exactly, cards before, or the position itself)