Here is something I came across here in the Café from last year. Someone wrote:

"Here is a probability with cards: After shuffling the cards any number of times, if you spread them, 9 times out of 10 there will always be a pair together. For instance, you may have two 7's together or two Kings, etc. Again, this is based on probability."

I was VERY intrigued by this, so I conducted 30 trials of thoroughly shuffling a deck of cards and spreading them out to see how many pairs were present. Just so you know, a pair is any two cards side-by-side of the same rank but NOT necessarily the same suit. Also, there were a few instances where three cards of the same rank were side-by-side. I counted this as two separate pairs. There was even one instance where four cards of the same rank were side-by-side! I counted this as three separate pairs.

Okay...here are the results of my experiment:

30 TRIALS

O pairs occurred 1 time
1 pair occurred 3 times
2 pairs occurred 6 times
3 pairs occurred 8 times
4 pairs occurred 4 times
5 pairs occurred 4 times
6 pairs occurred 3 times
7 pairs occurred 1 time

Based on this, I was trying to figure out some sort of mentalism-type effect to do with a spectator. Obviously, as they shuffle a deck of cards, I could predict that when spread out face-up, there are going to be a total of 3 pairs of cards in the spread (since 3 came up most frequently in my experiment). But it's not 100% guaranteed to come out that way, based on my results!! There may be four pairs! Or five! Or only two pairs!!

Anybody have any clever or creative ideas on how to use the principle I quoted above combined with the results of my experiment to produce a mind-blowing mentalism effect? Probability and statistics are not my best subjects!

Thanks!

There's an old gag similar to the above that's even more sure-fire: have someone name a card, and then show your gambling cheat shuffle chops by shuffling the deck and then showing that you've shuffled in such a skillful manner that there is now a pair of cards, one with the rank and one with the suit of the named card, next to each other, i.e. if they name the Three of Hearts, you show that there is a three next to a heart in the deck.

If you think about it for a bit, you have four threes in the deck and you have eight chances (either side of the three) to have a heart next to it. The probability that none of those eight cards is a heart is pretty low (less than 10%.)

[ Math types, I know I haven't adjusted for the possibility that the rank card could be at the beginning or the end, and my rough and ready calculation of 3/4 ^ 8 doesn't factor in that each succeeding card has fewer possibilities of not being a heart. Don't have calculator or pencil and paper nearby.]

Jack Shalom

That was a great reply, landmark! It really does work most of the time!

Does anybody else know anything peculiar about a randomly shuffled deck of cards? For instance, what is the probability of a group of four or five cards being in numerical order after shuffling (regardless of suits)? Is there even such a probability of that happening on a regular basis?

Any other ideas, comments, or suggestions would be appreciated!

This reminds me of the old trick in which a spectator is asked to name any two RANKS only -- Queen and three, for instance, or four and seven -- and then the deck is shuffled, the magician claiming to be rearranging the deck so that cards of these two named ranks will be found next to each other (no guarantee of what suits these cards will be). When you go through the deck after the shuffle, you find that (about 75% of the time) two cards of those two values will indeed be found together. Sometimes this will occur twice in the deck! It's just another surprising chance result.

One thing I've found by conducting several trial shuffles and spreading of the cards is that VERY OFTEN there will be only three pairs of cards of the same rank (not necessarily the same suit) in the spread (please see my initial post above).

Also, the shuffled spread has consistently yielded one set of five cards in a row of all the same color.

So two almost-surefire predictions would be:

1) There will only be three pairs of cards in the spread (same rank, not necessarily same suit) and

2) there will be five cards in a row of the same color.

Granted, these aren't going to happen 100% of the time, but it seems to "hit" more often than not. And being wrong part of the time is what lends an aura of credibility to our "powers."

Try it out yourself and see!

If you are interested in probability with pack of card you should read Persi Diaconis papers.

Where can one get these papers? Thanks!

Mathematical developments from the analysis of Riffle-shuffling. In A. Fuanou, M. Liebeck, J. (EDS) Groups Combinatorics and Geometry 73-97, World Scientific, N.J., 2003 M. Liebeck ed http://www-stat.stanford.edu/~cgates/PERSI/papers/Riffle.pdf

From shuffling cards to walking around the building. An introduction to Markov chain theory, Proc. Int. Congress, Berlin, vol. I, Plenary Lectures, 187-204

Riffle Shuffles, Cycles and Descents, with M. McGrath and Jim Pitman, Combinatorica vol. 15, 11-29

Trailing the dovetail shuffle to its lair, with David Bayer, Ann. Appl. Prob., vol. 2, 294-313

Analysis of Top to Random Shuffles, with Jim Fill and Jim Pitman, Combinatorics, Probability Computing, vol. 1, 135-155

Shuffling cards and stopping times, with D. Aldous, Amer. Math'l Monthly, vol. 93, 333-348

The mathematics of perfect shuffles, with R. L. Graham and W. M. Kantor, Advances in Applied Math., vol. 4, 175-196

You should also read this:

http://www.dartmouth.edu/~chance/teachin......Mann.pdf

I have to admit the math in those papers referenced above is way above my understanding.

Anyone able to translate what that might mean for magicians?

Two questions which I'm curious about: one of the conclusions seems to imply that 7 riffle shuffles are the optimal number to randomize a deck. Does that mean seven perfect i.e. faro shuffles?

If it does mean seven perfect shuffles, then it would seem counter-intuitive that the deck would now be randomized since it is common knowledge that eight faros bring a deck back to its original position. That would mean after 7 shuffles then only one more would be needed to completely unrandomize the deck.

Can anyone help to explicate this a little more?

Thanks,
Jack Shalom

No. it doesn't mean perfect shuffles. It means standard riffle shuffles

The riffle shuffle referred to in the papers is the following:
1) The deck is divided into two parts, call them left and right, that are approximately equal in number of cards. The less equal the division, the lower the chance of it happening.
2) The cards are released a little at a time from the bottom of the left and right portions. The number of cards released at a time from either portion is random and small. The probability of a large chunk of cards being dropped from either left or right is less than the prob. of a smaller chunk.
3) Chunks of cards fall from left and right at random.

Under the conditions above it was shown that it would take about 7 such shuffles to randomize a deck of 52 cards. Any more shuffles and it doesn't get significantly more random.

Note that the faro shuffle does not randomize the deck. Because exactly one card is dropped from alternate left and right halves the outcome of a faro is completely predictable. This is the basis of most of the faro shuffle effects.

To understand how the randomization in a riffle shuffle happens imagine that you have the ace through king of spades in order on the bottom of a deck. Give the deck one riffle shuffle and you haven't destroyed the order of the spades. If you look through the deck you will find them in ace through king order, with ace lowest in the deck and king located closest to the top. They may even retain the same relative order after the 2nd riffle shuffle. At the third shuffle the sequence of spades begins to be broken up, but there will still be some long runs of spades in sequential order. Such a bunch of cards is called a rising group and it was shown that as the riffle shuffles increase the size of the rising groups decreases and the number of them increases until about the 7th shuffle. After that, further shuffling doesn't increase the number of rising groups or decrease their size.

This can be the basis of a fun trick. Write down the order of a deck of cards and give it to someone with instruction to cut the deck into two approximately equal parts. Then have them remove and note one card from the top half and replace it anywhere in the bottom half. Then have them riffle shuffle the two halves twice. Then go through the deck a card at a time checking off the cards in the original order. You should find 4 groups of cards still in order, but one card will be out of order in its group -- the selected card. Obviously you'd want to do this by mail or over the phone, otherwise it would be pretty tedious (and obvious).

Hope this helped.

Dale Hoyt

Great summary Dale. Many Thanks,

Jack Shalom

I think it's imperative to do the shuffles before the selection procedure. It's _very_ possible to shuffle the selection directly into the correct position in the chain it came from, the way you described it. Great trick and it can easily stand three riffle shuffles with cuts between each shuffle, since it's enough to keep the chains of at least two cards in length.

/Tomas

TomasB, you posted a trick on this subject maybe a little over a year ago using two decks of cards with the turning of the cards in each deck over simultaneously. It was based on probability. Could you repost that here again? Thanks!

Here's the link to some ideas to use in combination with "The Frequent Miracle":

http://www.themagiccafe.com/forums/viewt......;start=0

Regards,

/Tomas

You da' man, TomasB!

Another principle related to the probability with cards is the called Kruskal Principle, discovered by the mathematician and physician Martin Kruskal. See on the web in order to know more details about that principle.

Best regards,

(Pedro)

I have a gospel routine that I thought of based on the above but I am not sure if it is the right place to post here so I am going to post it at the gospel magic section if anyone is interested.

Just posted it under good news:
Gospel routine with card probabilty system

Just tried Landmark's idea of having the value and suit next to each other. I chose the QH, and wouldn't you know it, at one point in the deck there was QS followed by 9H, making a match, and then immediately after was QD and AH, making a 4 card in a row double-whammy match! Simon Lovell talks a bit abouit probability in cards in his Card Shark tapes for anyone who was interested in looking into more of it.

Kevin