In the paper "Trailing the Dovetail Shuffle to its Lair", Bayer and Diaconis give a formula for showing how many times a deck of $N$ cards has to be riffle shuffled for the deck to be considered random. The formula they came up with is $\frac32 \log_2 N$, where $N$ is the number of cards in the deck. But what if your deck only contains $2$ cards, does the formula still hold , and what does it mean for a deck of just $2$ cards to be "random"?
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A deck of two cards being random means that either card is equally likely to be on top. If you start with a known order, say $1$ on top, and perform a riffle shuffle that is equally likely to be in or out, you have a random deck. An out shuffle leaves the deck as is and an in shuffle reverses the cards. Their formula says you need $\frac 32 \log_2(2)=\frac 32$ shuffles, not far from $1$
Ross Millikan
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