Interesting complex problem

[Scott_Baio]

Ok, I was presented with this problem today and it ended up being easier than I first thought it would be but still had some subtle nuances, one of which got me the first time I tried to solve it.

In the game cribbage, the best possible hand is having a J555 in your hand, and having the board (in cribbage the board is just the top card of the deck flipped up) be the fourth 5. Additionally, the five on the board has to be the same suit as the jack in your hand.

When you are dealt a hand in cribbage heads-up, you are dealth 5 cards, then discard 1. When you are dealt a hand with 3 players you are dealt 6 cards, then discard 2. The question is... How much does it hurt your odds of getting the best possible hand in cribbage when you go from 3 way to heads-up.

(this isn't an EV question, it's an odds question. So it doesn't matter that it gets equally harder for all players)

[Siegmund]

Small detail: you have it backwards - you get 6 cards heads up and 5 three- or four-handed.

If you are dealt 5 cards, you must be dealt 555JJ or 555Jx, and the case five has to be the starter. There are 24 ways to get 555JJ and 704 ways to get 555Jx, so with 5 cards, it is 1/47 * 728/52C5 = 1/167790.

If you are dealt 6 cards, you can be dealt 555JJJ, 555JJx, or 555Jxx, 16 1056 or 15136 ways respectively, and the case five has to be the starter.
1/46 * 16208/52C6 ~ 1 in 57780. Almost exactly three times as likely.

(Divide the above by four, to force the jack and starter to be of the same suit - the relative probability remains the same.)

I have never held a perfect hand, myself, over the course of several years of playing cribbage. Seen someone else get dealt 5555 and turn up a face card for 28 once.