Some Cribbage Math (and Simulation)
Before jumping into the text, here’s a refresher on the scoring rules of cribbage (specifically, 6-card 2-player cribbage). The game is a race to 121 points, starting from 0. These are not the full rules for playing the game, just the rules for scoring a hand.
First, each player is dealt six cards. Each must discard 2 cards to the crib, a bonus hand that is received by the dealer at the end of the current round (dealer changes each round). after both players discard to the crib, a card (“the cut”) will be flipped upright from the deck, and considered an addition to both players’ hands. Hands will be scored in the following ways:
Fifteens: Every distinct subset of cards such that their sum (card ranks, and 10 for all face cards) is exactly 15 will be awarded 2 points.
Pairs: Every distinct pair of same-rank cards will be awarded 2 points.
Runs: Every distinct sequence of consecutive card ranks will be awarded exactly as many points as the length of the sequence, provided that the length is at least 3 cards. Sequences cannot be split into smaller subsequences for additional scoring.
Nobs: If a Jack in your hand is of the same suit as the cut-card, that player will be awarded 1 point.
Nibs: If the cut-card is a Jack, the dealer will be awarded 2 points.
Flush: If all 4 cards in-hand (before the cut-card) are of the same suit, that player will be awarded 4 points. If the cut-card is also of the same suit, that player will be awarded an additional point, for 5 total.
The total score of a hand is the sum of each of the above scoring methods. The crib is subject to the same scoring rules as any other hand, except that the crib can only score a flush if all 5 cards (including the cut-card) are of the same suit.
Introduction
Cribbage is a game about probabilistic decision making. It can often be hard to determine what the optimal cards to throw to the crib are, given that we aren’t computers, and the differences in expected value based on what you throw can be very minor.
Consider the following hand (with S = spades, C = clubs, H = hearts, and D = diamonds). It’s your crib, and it’s time for you to throw two cards:
Ace (S), Ace (H), 4 (S), 6 (D), 7 (D), 8 (C)
According to some calculators, the best cards to throw to the crib in this scenario are Ace(S)-4(S) for an expected 13.90 points or Ace(H)-4(S) for an expected 13.84 points. This is considering expected points from both hand + crib. That minor discrepancy between the two options, with the first being ~0.06 points better (on average), comes entirely from the suits of the cards, as both options throw the same ranks into the crib. This difference is significant because the first option allows for flushes in the crib whereas the second does not.
One might naturally ask, just how much does allowing flushes in the crib affect average crib value? In other words, how much does throwing “on-suit” versus “off-suit” cards matter?
EV from Pairs
The first thing to note is that the combinatorial point structure that exists is different conditioned on whether or not there exists a flush in the crib. The biggest difference is that the existence of a flush (all 5 cards having the same suit) necessitates the fact that there exist no pairs in the crib. So, we’ll need to attempt to find the average number of points lost from a lack of pairs, which will be offset by 5 points gained from having a flush.
We’ll use the fact that the expected value (EV) from pairs will be the probability of attaining each combination of cards leading to each type of “pairing” scenario in-hand, multiplied by the value of such a scenario (in points). We use 5 cards in a randomized hand instead of 6. This is because in a standard game, we will be dealt 6, discard 2 to the crib, and be given 1 more card via the cut, for a net total of 5 cards for scoring.
This is not a perfect estimate; real players don’t play with randomized hands, they will tactically preserve points for themselves or deny points from their opponent when possible. If anything, this is likely an underestimate, as a real player will see a total of 7 cards during their turn (6 dealt and 1 cut card), not just 5.
All together, we have that the expected number of points gained from pairs is approximately:
\[ 2\Big( \frac{1760}{4165} + 2\frac{198}{4165} + 3\frac{88}{4165} + 4\frac{6}{4165} + 6\frac{1}{4165} \Big) = \frac{20}{17} \]
or about 1.176 points on average. These are average points that are importantly lost from our randomized crib, if our crib contains a flush.
Theoretical EV Difference for Flush vs. Non-Flush-Allowed Cribs
Now that we know roughly how many expected points will be lost from cribs with flushes, we can estimate the value of a flush-allowed vs. non-flush-allowed crib. Let \(p\) be the EV of a crib in which we throw two cards of differing suits (flush is impossible). Let \(q\) be the EV of a crib in which we throw two cards of the same suit (flush is possible). Then, \(q - p\) would be the gain in EV from throwing two on-suit cards vs throwing two off-suit cards.
We can approximate \(q\) with a weighted average:
\[ q \approx \frac{\Big(\binom{50}{3} - \binom{11}{3}\Big)p}{\binom{50}{3}} + \frac{\binom{11}{3}(p+5-\frac{20}{17})}{\binom{50}{3}} \]
The first term comes from the fact that there will be \(\binom{50}{3} - \binom{11}{3}\) ways to “finish out” the crib in a way that does not result in a flush, given that two cards were tossed that were of the same suit, each of which scoring an average of \(p\) points. The \(\binom{11}{3}\) comes from the other 11 cards of the same suit remaining in the deck after the two discards.
The second term comes from the fact that there will be \(\binom{11}{3}\) ways to “finish out” the crib in way that does result in a flush, each of which scoring an average of \((p + 5 - \frac{20}{17})\) points, owing to the fact that the flush will add 5 points on top of \(p\) while removing that \(\frac{20}{17}\) from earlier. Then, we take the following and simplify:
\[ q - p \approx \frac{\Big(\binom{50}{3} - \binom{11}{3}\Big)p}{\binom{50}{3}} + \frac{\binom{11}{3}(p+5-\frac{20}{17})}{\binom{50}{3}} - p \] \[ q - p \approx \frac{\binom{50}{3}p - \binom{11}{3}p + \binom{11}{3}p + \binom{11}{3}5 - \binom{11}{3}\frac{20}{17} - \binom{50}{3}p}{\binom{50}{3}} \] \[ q - p \approx \frac{\binom{11}{3}5 - \binom{11}{3}\frac{20}{17}}{\binom{50}{3}} = \frac{429}{13328} \]
In other words, our estimated gain in EV from throwing two on-suit vs off-suit cards to the crib is about \(\frac{429}{13328} \approx 0.032\) points.
Experimental EV Difference
We can actually experimentally estimate this difference in EV via simulation. After writing out all of the necessary scoring functions and scoring 10,000 randomized crib hands for all 1326 possible pairs of discards, I found an average difference between on-suit and off-suit discards (for card pairs of equivalent rank sets) of about 0.06458828 points.
My best guess as to why there is a discrepancy (0.032 theoretical vs. 0.064 experimental EV change estimates) is that the available pool of cards to pick from, when necessitating flush-allowing vs non-flush-allowing hands, is slightly different. There would likely be greater run potential for same-suit discards; the theoretical model assumes the same baseline points for a flush vs non-flush crib, differing only by points added by the flush and removed by the lack of pairs. In reality, the point structures of runs and fifteens would also be impacted by forcing a flush in the crib. We could probably achieve a more accurate estimate with a more robust model that captures these EV changes in runs and fifteens when forcing flushes, though the math gets very annoying very quickly.
At the end of the day, however, both the theoretical and experimental results agree in suggesting that rank matters far, far more than suit when it comes to discarding to the crib.
Other Simulation Explorations
In addition to verifying a combinatorial estimate, the scoring function allows us to analyze other aspects of the game as well. Below is a heatmap of average randomized crib values, given various discards from the hand.
Though this heatmap is again subject to the limitations of randomized cribs, we can still observe interesting patterns.
For instance, as expected, even a single 5 thrown to the crib results in a high crib EV. Some red-orange spikes can be seen through the entire heatmap on the row and column corresponding to 5, and throwing two 5s results in by far the highest expected crib EV out of every possible pair of cards to throw.
There exists a cluster of higher crib EVs for pairs of discards such that both discard ranks are \(\le 5\), likely due to the potential of those discards to make 15s with any 10 or face-card that ends up in the crib. Interestingly, though the Ace and the King are equally rare for presence in runs, the fact that an Ace can be combined with other cards to make many 15s is likely why contributing an Ace to the crib will generally result in higher EV than a King.
Any thrown pair seems to do well in the crib, as can be seen by the red line down the diagonal. Likewise, any guaranteed 15 (7 + 8, 6 + 9, etc.) also does well. Guaranteeing some points appears to do better, on average, than cards that “might” do well in the crib.
In this histogram, we can observe that most discarded pairs yield an EV of roughly 4 points in a randomized crib, with essentially zero in the 8-ish point range, and a few discarded pairs in the 9-ish point range. Neat!