Many of you are probably familiar with "Penney's Game", where you have someone choose a sequence of 3 coin flips (such as HHT, for example), then you choose a sequence a coin flips, a coin is flipped, and whoever's sequence comes up first is the winner. The scam is that, regardless of what sequence the other person chooses, it's always possible for you to choose a more likely sequence.

For those not familiar with this classic betcha, it's a percentage bet. You can lose, but over the long run, you'll win more often than you lose. For a more detailed description of the classic version of "Penney's Game", watch this James Grime video or this Scam School video.

There's a variation now that uses playing card colors (red or black) instead of coin flips (heads or tails). The probabilities are different, because coin flips are independent of each other, while cards being removed from the deck are dependent (the colors of the removed cards affect the distribution of the colors remaining in the deck). It's also different because draws are possible (it's possible to make it through the entire deck without either player winning). Fortunately, the winning strategy is still the same! You can read about the card version, and the probabilities involved in the post The Humble-Nishiyama Randomness Game.

A Monte Carlo computer simulation was used to discover the results listed in the post, but there are agonizingly few details about the exact process used. Did the computer randomly order the cards each time, or did it simulate a random riffle shuffle? Why only 1,000 times? Even my 4-year-old laptop can run millions of Monte Carlo simulations quickly.

Any thoughts on the card version of Penney's Game?

Hi, Scott. First, thanks for all your contributions. They are always enlightening. The version I use for Penney's Game is in The Card Magic of Nick Trost, page 89, under the title "The Penney Paradox". Trost employs cards and a very clear way of tallying the results. I enjoy reading about probabilities but am not an expert, to say the least. I hadn't realized there can be ties. Do you have the probabilities for that? Despite the ties (which have not happened to me so far, but now I'm ready), please do try the Trost version. Spectators (marks?) can follow it better.

Thanks, federico!

According to the link I gave above, here are the probabilities after running 1000 simulations:

BBB vs RBB – RBB wins 995 times, 4 draws, BBB wins once
BBR vs RBB – RBB wins 930 times, 40 draws, BBR wins 30 times
BRB vs BBR – BBR wins 805 times, 79 draws, RBR wins 116 times
RBB vs RRB – RRB wins 890 times, 68 draws, RBB wins 42 times
BRR vs BBR – BBR wins 872 times, 65 draws, BRR wins 63 times
RBR vs RRB – RRB wins 792 times, 85 draws, RBR wins 123 times
RRB vs BRR – BRR wins 922 times, 51 draws, RRB wins 27 times
RRR vs BRR – BRR wins 988 times, 6 draws, RRR wins 6 times

I knew magicians (I even should've guessed Nick Trost in particular) would've come up with card version long before mathematicians, but we still needed the mathematicians to work out the probabilities.

I submitted this variation to Scam School when I first ran across it, and it aired as last week's episode! You can see it in action at: https://www.youtube.com/watch?v=fCz8xYLGmFY

Excellent, thanks Scott.

I had a blast trying this on a couple friends that played against me. We went trough the deck a few times, but we bet on every single outcome instead of betting on the final outcome (who's got the more tricks). It allowed for them to be ahead once or twice before losing miserably

I am going to add this to my repertoire.

Claudio, I like that variation of it! It's one thing to see a loss when the tricks are coming, but losing the money each time really makes them feel the loss!

I actually thought about this a bit further and came up with a couple of ideas.

For instance, after one round (i.e. the full pack has been dealt) or two, you could offer your assistant to switch the rules so that the winner is the one who does not get a match. "That should play in your favour ..." you'd tell your spec, jokingly. If the challenge is accepted, all you have to do is, once the spectator has displayed his combination, ensure your combination is a loosing one by reversing the combination rules. It reads perverse, and I think it is

You could also play with two packs at once (either shuffled together or separated.) I don't know the maths, but it should even be more favourable to the performer. I'll have to run some simulations.

You could allow the spectator to "stay" or call for a new combination at the half-way point.

Rather than displaying cards, using six poker chips-each red on one side and black on the other, as suggested by Nick Trost, is an excellent idea. The advantage is that the spectator does not have to select his combination first as you can both play with the chips to select a combination, while in fact you adjust yours to the spectator's.

It's possible to implement the above idea with cards of course. Both players start with 3R and 3B cards to choose from.

I have a few more ideas, but I'll stop the rambling

I am thinking of developing an Android application on this. I already have a couple of ideas.

I am not familiar with the Monte Carlo game used for simulation. I googled it and it seems to be a simplistic solitaire game. I am not sure why it was used for simulations. Any ideas why?