Blaming bad luck

While playing cribbage on Board Game Arena, my opponent complained about a 2 point hand with the crib:

Screenshot 2023-08-27 at 8.26.33 PM

They claimed this was “almost mathematically impossible”. The problem with that thesis is the odds are pretty well defined:

Score Number of hand-crib pairs (out of 2,317,817,502,000) Percentage of hand-crib pairs to 6 decimal places Percentage of hand-crib pairs at least as high
0 14,485,964,652 0.624983 100
1 3,051,673,908 0.131662 99.375017
2 80,817,415,668 3.486789 99.243356

So just under 3.5% of hand-crib pairs score exactly 2. Even a combined score of 0 happens from time to time. Now these statistics assume the two cards passed from the initial deal to the crib are random, which isn’t the way cribbage is played. If you have the crib and a pair in your opening hand, you’d be unlikely to split the pair. Same is true for 15s, runs and flushes. A random play might make discards that increase the odds of getting a zero-point crib. So the odds of getting 2 or less are actually somewhat lower for people actually playing cribbage.

My opponent misplayed this hand rather badly. What would you discard to your crib after being dealt these cards?

A :clubs: 2 :clubs: 4 :hearts: 5 :spades: 7 :hearts: J :diamonds:

Looking at it, I’d probably discard the 2 and 7 which gives me at least 4 points in my hand. Checking an online tool for analyzing hands, I found it was not only the best discard, but also guaranteed 6 points. Why? Because every possible card that can be flipped would add at least two points to my hand or crib. Obviously I could get a pair with any A, 2, 4, 5, 7 or J. Or I could get 15 with a 3, 6, 8, 9, 10, Q or K. So that means no matter which card gets flipped, this hand plus the crib will be worth at least 6.

But my opponent discarded the 7 and J! That separated the two guaranteed 15s. The hand will still be worth at least 2 because any card that’s flipped will either create a pair or 15. There’s no real chance this discard will result in a higher score either. The tool says discarding 7-J will score an average of 7.3 points. The best play (2-7) averages out to 11.2.

We tend to give ourselves credit for superior skill when we do well and excuse ourselves with inferior luck when we lose.