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In two-player cribbage, when the pone is dealt a hand that includes a run and a 5+pair (e.g. 3,4,5,6,Q,Q) is it better to keep the run (3,4,5,6) or the 5+pair (_,5,Q,Q)?

It seems like there should be a statistically optimal play here.

In the specific example above, either choice leaves the pone 6 points, but the run has a chance of improving to a double run (12 out of the remaining 46 cards) and is improved by 24 other cards (2s, 7s, Ts, Js, Qs, Ks). The main downside of that play is it concedes a pair to the dealer. On the other hand, the 5+pair can improve through any 5, Q, or ten-value card. If the connecting 6 is kept, it can also improve through a 4 or 7. Overall, there are 24 cards that improve that hand. It also concedes a 3,4 to the dealer - which seems less likely to give the dealer points.

My intuition is to keep the run, but a mathematically sound analysis of expected value would be nice.

P.S. For the purposes of this question, we can ignore the exact position of the players, and assume it's somewhere mid game (as opposed to really close to the end where pegging considerations become more important).

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If your opponent had the crib, I'd throw him a 3 and a Q –  Sam I am Feb 1 '13 at 20:06
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2 Answers

up vote 5 down vote accepted

I'm pretty sure the 'optimal' play is neither of the ones you mentioned! Keeping 4,5,6,Q counts 7, gives you essentially equal opportunity for another 2 points off of any face card (you lose out on four extra points if one of the other two Qs comes, but that's only a 1 in 7 chance), still earns you all but two of the points you'd have had from 3,4,5,6 if you hit a double-run (though it slightly lessens the odds of it by cutting out the 3 as a double-run card), and gives your opponent just about the worst of the possible combinations for their crib (3Q).

To match Basicpract's analysis: you score two extra points off of any Ace (4), one point off of any trey (3), seven points off of a four or six (6), 9 points off of a five (3), one point off of a seven (4), two points off of a 9 (4) four points off of a Q (2), and two points off of any other face card (12). This is 8 + 3 + 42 + 27 + 4 + 8 + 8 + 24 = 124 points off of the remaining 46 cards, for an average of more than two and a half points improvement and an average overall hand value of 9.7 points, compared to the 8 points from keeping 45QQ and the 9.61 points from keeping 3456 (but with a much worse crib to your opponent).

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Interestingly, a number of online Cribbage hand analyzers recommend this exact play. Nice work. –  LBushkin Feb 2 '13 at 16:57
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So using some statistics, lets see what happens!

Keeping 45QQ is the best chance if you want to keep the pair. Aces(4): +2; Twos(4): +0; Threes(3): +3; Fours(3): +2; Fives(3): +6; Sixes(3): +5; Sevens(4): +0; Eights(4): +0; Nines(4): +0; Tens(4): +2; Jacks(4): +2; Queens(2): +6; Kings(4): +2.

There are 46 cards left that could be cut from, 30 of them give you points. 19 give you 2 points, 3 give three points, 3 give 5 points, 5 give you six points. The average number of points you will gain then is 2 points (92/46).

Keeping 3,4,5,6 is much better! Aces(4): +2; Twos(4): +3; Threes(3): +6; Fours(3): +8; Fives(3): +8; Sixes(3): +8; Sevens(4): +3; Eights(4): +2; Nines(4): +2; Tens(4): +2; Jacks(4): +2; Queens(2): +2; Kings(4): +2.

Of the 46 remaining cards, all of them give you points. 26 cards give you 2 points, 8 cards give you 3 points, 3 cards give you six points, and 9 cards give you 9 points. On average than that is 3.61 points. Clearly superior.

If your run is not with 456, or containing 7,8, then the point values go down quite a bit. Also runs beginning with A or ending in K have less potential value, as the straight only goes one way. If the par is Jacks, that is a 50% chance of getting one more point.

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I lost you when you said throwing your opponent two queens in his crib was much better... Your average of 3.61 points, minus the two points you are giving your opponent, means you only gain 1.61 points on average, while the first option averages 2. Clearly superior. –  corsiKa Feb 1 '13 at 16:04
    
This looks like a pretty good start. Shouldn't we also look at the opportunity for the dealer to get points from each option? I agree with corsiKa that the two points (Q,Q) should count against the expected value of the hand - but it may actually be worse/better depending on the potential for the dealer. –  LBushkin Feb 1 '13 at 18:25
    
Hmm, I got so caught up with the numbers, to be honest, I forgot about the crib completely. I think it is fair to say that with the opponent's crib taken into account, throwing a pair into the opponents crib will help them, but I am not sure yet what average crib values for a QQxx crib and a 36xx crib are. I will spend some time (probably more time than I should) to see if I cannot figure those out. The question boils down to "is a QQxx crib 1.61+ points better than a 36xx crib", and corsiKa, looking at that now it seems quite likely, but I will try to get some numbers! –  Basicpract Feb 1 '13 at 18:43
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