A chi-square test of the Euchre shuffle

By Marya · Updated by Holger in September 2026

If the shuffle of your Euchre deck is truly random, then the chance that the up card is any rank you might pick is 1 in 6. Why is that, you may ask? Well, there are 6 possible ranks (9, 10, Jack, Queen, King, and Ace). There's no reason for one rank to turn up as the up card more often than another. So, for example, the chance that the up card is a Jack when you deal a hand is just 1 in 6.

If you want to check whether your shuffle is fair, the rank of the up card is one of the easier things to test. What you want to do is shuffle the cards many, many times. After each shuffle, deal out the cards and count how many times each rank turned up as the up card. For example, after dealing 96 hands, you might have seen an Ace as the up card in 17 of them, a Jack in 20, and so on. Once you have these numbers, you may not be quite sure what to do with them. You might think you'd find each rank as the up card exactly 16 times (6 ranks times 16 is 96), but that's unrealistic. Since the cards are random, you wouldn't expect each rank to show up exactly 1/6 of the time. So how do you decide from your numbers whether the shuffle is random?

If you think about it, the rank of the up card is like the roll of a 6-sided die. There's a Stack Exchange page that describes how to run a chi-square test to check whether an n-sided die is fair. There's a bit of math on that page. You can check it out or read the Wikipedia page on the same topic if you want more detail. The general idea is that you work out a χ² value for each side of your die (1 to 6 for a 6-sided die), add up all the χ² values, and compare the total with the "upper-tail critical values of χ² distributions" in a handy table.

I did this for 226,864 Euchre hands collected over a few weeks in April 2018. The χ² value came to about 5.5. There are 5 degrees of freedom, because there are 6 possible ranks. According to the χ² test, if the shuffle is fair, χ² will be less than 9.236 in 90% of all tests and less than 20.515 in 99.9% of all tests. Since my χ² value came in at 5.5, I'm very confident that the shuffle at World of Card Games is fair, random, and good enough! It uses an algorithm called Fisher-Yates.