What happens if I set up for a "29 hand" and miss?

Question

Inspired by an answer to "Why does a run of 4 in Cribbage score only 4?".

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The best possible hand in cribbage is 29, which involves the Jack that counts for nobs and all of the 5s. Now although hands with lots of fives are always good, often retaining just one or two will only result in a mediocre score where you count a few 15s, a pair, and are done.
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For example if you're drawing for the best possible hand failing to get that last 5 as the up card will cost you 14 points, as you'll only score 15-8, a pair royal for 6 and 1 for his nobs for 15.

Note: I think the count given is a bit off, or assumes something not explicitly stated - 555J on its own, with the J not matching any of the 5s - scores 14. Four 15s (8), plus trips (6). So, any up-card that doesn't add to that score in any way costs you 15.

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As stated above, the best possible hand in cribbage is an extremely rare case. It first requires a player to be set up with a hand that happens rarely on its own (555J, all different suits). Then, it requires the up-card to be one specific card - the 5 matching the in-hand J.

Playing for this scenario, when dealt the proper hand, could be a very risky proposition. There's about 15 points of difference between the minimum and maximum possible values of this hand, depending on the up-card. Though the minimum score of 14 is still a fairly decent hand, there's many ways that the opponent could easily out-score this hand.

Additionally, keeping this hand sets the player up to have points pegged against him which might be more easily avoided with a different hand selection. If the player is forced to play a 5 when the count is zero, there's a very strong possibility their opponent has something to make 15 with.

Of course, the risk/reward scenario isn't just a matter of "I might score 14 or I might score 29". There's still a number of possibilities in between. So, what are the general odds that the final hand score might still be favorable? What possibilities exist between 14 and 29 here, and which are most likely?

Answer 1

Indeed, playing for 29 can be a very risky proposition - especially if you're in a late stage of the game and stand a good chance of losing if your hand scores no more than 14.

To simplify, I'm setting aside any human influence here. Yes, there are two other cards known to the player, as well as six cards known and controlled by the opponent. However, none of these are going back into the deck and no further cards are being drawn.

At best, a player might be able to use the two other cards in his hand to adjust his expectations of what the up-card could be. Of course, there's no chance at nailing 29 if you have all four 5s dealt to you along with a J and something else. But if your extra two cards each have a face value of 9 or lower, the odds that the up-card will substantially improve your score just went up a notch.

Still, setting aside all factors except the 4 mandatory cards in the starter hand simplifies calculations and gives a relatively solid baseline for our expectations.

Once you take out the rainbow hand of three 5s and a J, you have 48 cards left in the deck. Of these, the possible up-card scenarios are:

• The 5 that matches the in-hand J (29 points, 2%) - This is the only way to score 29 and maximize this hand. And there's only one card that can do it.
• Any remaining J (22 points, 6.25%) - This adds three more 15s and a pair to your base, for an 8-point boost. Counting out just the given base hand, there's three of these left in the deck.
• T/Q/K matching the in-hand J (21 points, 6.25%) - Again, giving you three more 15s. Add on the knob and it's a 7-point total boost. Also, like the Jacks, there's only three ways to do this.
• Any other T/Q/K (20 points, 18.75%) - Not matching the in-hand J, but still having face value of 10, means we lose the knob/pair points but keep the 15s. There's nine ways to do this though, so our odds here are relatively favorable in comparison to the previous scenarios.

Totaling the above gives us a one-in-three chance of scoring 20 or better. With a minimum possible score of 14, that's still a very nice hand!

However, as you'll see below, two-thirds of the possible scenarios will only bump your score by one point at most - and the majority of those don't add to it at all. If scoring 16 or more is absolutely crucial in the round where you get this hand, you might want to put off your dreams of hitting 29 for another day.

• Any other card matching the in-hand J (15 points, 16.67%) - Anything other than 5/T/Q/K, which matches the in-hand J, only gives us one additional point for knobs. No extra 15s or pairs. This is where things start looking relatively bleak, since there's eight ways to do it.
• Any card not categorized above (14 points, 50.00%) - This is the bare-minimum possible score with a hand that might otherwise score 29. We've got the three 15s for pairing each 5 with the J (6), the one 15 by adding all the 5s (+2), and a three-card match between the 5s (+6). Any up-card that's not a T/J/Q/K, and doesn't match the in-hand J, will not add to this. Having 24 ways to do this, out of 48 possible up-cards, gives us coin-flip odds of being stuck with the minimum score for this hand.