Odds Of Opposing Runs In Trump

[MattK1]

Hello. I am a rank beginner and was playing pinochle last night for only the 3rd time. My fellow players were my girlfriend, her sister and her brother-in-law. We are all in our 50's and they have been active players for decades. Each of them have played tens of thousands of hands. Last night, a hand occurred that astounded my partner and our opponents and I'm curious about the odds/probability of this occurrence.

The details : We were playing single deck, 4-handed. My partner dealt. Her brother-in-law passed. I bid the minimum (21) and was outbid by the fourth player. The bid was eventually taken by my opponent at 27. (Again, I am BRAND NEW to the game, so although I realized I had a very strong hand, I did not understand the importance of winning the bid.) I had a Run with a 9 in Hearts and Kings Around. Here's where it gets strange and the big question arose. My opponent declares Hearts as trump and lays down a Run with a 9 in Hearts! What are the chances of this happening? 

And, I know it doesn't matter but we scored 42 points in the hand and STILL lost the game. Oh well....

Thanks in advance for any input.

MattK1

[marya]

Hi MattK1, I only play Double Deck but it would be interesting to hear from anyone who knows the probability on this one! - Marya

[ToreadorElder]

If you're still around...?

Don't want to run exact math, but...incredibly, vanishingly small. The 12 cards of that suit can be dealt to any of the 4 players equally...roughly speaking, it gives 4 ^ 12 ways it can happen. For both players to have th 6 card run:

First ace can go to anyone...4 players.
Second ace can go to anyone else...3 players.
If first ace goes to player 3, and second ace goes to player 1, it's the same as first ace to player 1 and second ace to player 3, so divide by 2.
First 10 can go to 2 players; second 10 can only go to 1 player.
First K can go to 2 players, second to 1. And so on.

So we have 4 * 3 / 2 * 2^5 (for each subsequent rank). So we have 6 * 32, or 192 ways it can happen...out of the 16 MILLION possible distributions.