I was working on an optional math problem given by my math professor John Golden at GVSU. The assignment was to analyze the probability of one of our favorite games. I chose to analyze the game Cribbage, as my grandpa taught me how to play and I’ve many fond memories of playing with him on warm summer days. For those of you who don’t know how to play cribbage, I’ll give you a quick run-down of the rules that will be relevant for our task here, which is to find the probability of obtaining a perfect hand in a two person game of cribbage.
In Cribbage each player is dealt 6 cards. They then choose two cards to discard into the “crib” which is essentially and extra hand for the dealer. After this is done, the remaining deck is cut and the top card flipped over. This card will be included in both players hands, as well as the crib, giving each player a hand of five cards (two hands for the dealer including the crib). One of the main ways points are scored is by grouping some or all of your cards to sum to 15, e.g. 6+9, 7+8, 2+5+8, 5+King. The last sum holds because in cribbage all face cards have a value of 10. Each 15 you make is worth two points. Another way to score is by having pairs of the same card, for example two fives. Each pair is worth two points, and sets of three cards the same counts as three pairs, thus is worth 6 points. Another seemingly random way to score points is by having the Jack of the same suite as the card which has been cut, this is called nobs and it is worth one point.
Now that we all know the ways to score the points in the perfect hand (there are a couple other ways to score which I have chosen to leave out here), we are ready to talk about what the perfect hand is. The perfect hand consists of three 5’s and a Jack which is a different suite than all three fives. Then the fourth 5 must be cut to be the fifth card in the hand. The perfect hand is worth 29 points but it will be helpful to see where all those points come from. Four 15’s can be made with the Jack and each of the 5’s, giving us 8 points. Four more 15’s can be made of three 5’s, giving us a total of 16 points. Because we have four 5’s, we have six pairs, each worth two points, giving us a new total of 28. Of course we cant forget nobs, giving us a grand total of 29 points.
I’ve known about this hand for some time now and have wondered how likely it is to occur. I decided to try to find out on my own. I did so by creating a table which mapped out the dealing of the cards. I had a column for who the card was being dealt to, O for opponent and D for dealer, one for the desired card, and a final column with the probability of dealing the desired card. Here’s a link to the table I created (forgive the awkward link title): Card goes to Card needed probability.
After thinking through each card dealt and its associated probability, I figured the hard work was done. I simply multiplied all the probabilities together to get the probability of all of the events occurring at once. The answer I got was 7.2×10^(-8) which is about 7.2 out of a hundred million. This answer seemed to be reasonable to me, knowing that obtaining the perfect hand is something that rarely occurs. However, I consulted wikipedia for confirmation and found a bit of a discrepancy (98%). The odds on the web were about 1 in 216580, or 4.617×10^(-6). I found similar numbers on other sites as well as one claiming to see such numbers “experimentally” in cribbage tournaments. How was I off by two orders of magnitude?
I did some reworking with a friend ignoring the dealers cards. We did this because in one of the calculations online (which we had a hard time following) involved multiplying by 1/46, which we deduced was associated with the probability of cutting the last 5. After going through the calculations as if there was only one person playing (such a sad game), we came up with the probability of 7.695×10^(-8), which was only slightly better than before. We then had a huge “ahaa!” moment when we realized that there were four perfect hands, i.e. one for each Jack. We multiplied our answer by a measly 4 and got 3.078×10^(-7).
We were getting much better, but an order of magnitude off is still quite a bit. After banging our heads against the wall for a half hour or so, we came to the realization that we were forcing the perfect hand to be dealt in the first four cards of the six. We did some thinking and came to the conclusion that there are in fact 15 different ways (6 choose 4) different ways to deal each perfect hand, each having the same probability. After multiplying our last probability by 15 we got 4.617×10^(-6)! We were able to match the probability given by wikepedia on our own, which excited us immensely.
We were both a little off put though that the accepted probability of obtaining the perfect hand was calculated in a situation where one person sits down and deals themselves 6 cards, cuts the deck and then flips the last card. After all, that is not how cribbage is really played. We took what we learned trying to match the probability posted on the internet and applied it to my original probability. After multiplying by a factor of 60, the probability we obtained was 4.32×10^(-6). Both my friend and I feel that this is a better approximation of the probability of obtaining a perfect hand in a two handed game of cribbage.
In doing this I learned that one should never accept what they see without skepticism. If one is indeed skeptical enough, they should try to work out the problem on their own. They may just find out that the accepted answer doesn’t fit their standards. Also, a lot is learned by working things out, opposed to taking the answers for granted.