The dice game craps is played by repeatedly rolling two six-sided dice, and making various wagers on the outcomes of the sequence of rolls. The details of the wagers don’t concern us here– instead, let’s consider just one particular example scenario involving a common wager called the “pass line bet”:

Suppose that we have just rolled a 4 (which, by the way, occurs with probability 3/36 on any single roll). Having thus established 4 as “the point,” our objective now is to roll a 4 *again*, rolling repeatedly if necessary… but if we ever roll a 7 (which, by the way, occurs with probability 6/36 on any single roll), then we lose. If and once we roll a 4, we win.

To summarize: we are trying to roll a 4 (with probability 3/36). If we roll anything else except a 7 (where “rolling anything else except 7” has probability 27/36), we continue rolling.

So, here is a puzzle, let’s call it **Problem 1:** how long does it take on average to *win*? More precisely, what is the expected length of a *winning* sequence of rolls, i.e., where we never roll a 7?

**Thoughts**

This problem is not new, nor is it even particularly sophisticated. But it is very similar to a problem that circulated a few years ago, and that generated some interesting discussion. Here is that earlier problem, let’s call it **Problem Zero:**

Roll a single fair six-sided die repeatedly until you roll a 6. What is the expected number of rolls required, given that we observe that all rolls are *even*?

My motivation for this post is two-fold. First, this is a sort of pedagogical thought experiment. Problem Zero has already been shown in the wild to be dangerously non-intuitive. Problem 1 is *the same problem*— that is, it is essentially equivalent to Problem Zero, just dressed up with different values for the single-trial probabilities. But is Problem 1 inherently easier, less “tricky?” And if so, why? Is it because the numbers are different, or is it that the problem is cast in a more concrete setting as a casino game, etc.?

I don’t know if these problems are actually equally “tricky” or not. But at least in the case of the known trickiness of Problem Zero, I have a theory. Both of these problems are like many others that I have enjoyed discussing here in the past, in that they are *mathematical* problems… but of a sort that a student may approach not just with pencil and paper, but also with a computer, even if only as an initial exploratory tool.

Which brings me to my theory, based on observation of past debate of Problem Zero. We can begin tackling this problem with pencil and paper, or by writing a simulation… and I suspect that, in this case, starting with a simulation makes it much *harder* to come up with a (the?) *wrong* answer.

Riffle shuffle a deck of cards once. What is the probability that the original *top* card ends up on the *bottom* of the shuffled deck (or vice versa)? This is very unlikely… but suppose that we shuffle again… and again, etc. How many shuffles on average are required until we first see the original top card moved to the bottom of the deck? The motivation for this post is to capture my notes on this problem, which turns out to have a very nice solution.

**Approach**

To begin, we use the Gilbert-Shannon-Reeds (GSR) model of a riffle shuffle, that captures– in a realistic but mathematically tractable way– the random imperfections in how most non-expert humans shuffle cards: by cutting the deck roughly in half, then interleaving the cards back together from the two halves, with some clumping.

There are several different characterizations of the GSR model, all yielding the same probability distribution of possible permutations of the cards in the deck. For our purpose, the most convenient way to model a single random GSR shuffle of a deck with cards is as a uniformly random string of bits (imagine flipping a fair coin times). The *number* of zero bits indicates how many cards to cut from the top of the deck, and the *positions* of the zero bits indicate how those top cards are interleaved with the cards from the bottom part of the deck (represented by the one bits). The following Python code illustrates how this works:

import numpy as np def riffle_shuffle(cards, rng=np.random.default_rng()): bits = rng.integers(0, 2, len(cards)) return [cards[k] for k in np.argsort(np.argsort(bits, kind='stable'))] print(riffle_shuffle(range(52)))

This representation of a riffle shuffle as a bit string makes it easy to calculate probabilities of various types of shuffles. For example, of all possible equally likely GSR shuffles, how many move the card from position down to position , where ? This number is equal to the number of bit strings with at least zeros (since the card we want to move *down* must be in the *top* half of the cut), and exactly ones before the -th zero. That is,

It’s an exercise for the reader to show that we can extend this approach to also count shuffles that move a card *up* in the deck (), or that *fix* a card () so that it remains in its original position, with the result being a transition matrix given by

where indicates the probability that a single GSR shuffle will move the card in position to position .

At this point, we have what we need: consider a Markov chain with states corresponding to the current position of a particular distinguished card in the deck. If this distinguished card starts in position, say, (i.e., the top card of the deck), then our initial state distribution is the -th basis (row) vector, and we can track the distribution of possible future positions of the card after shuffles as .

But we originally asked for the *average* number of shuffles needed to move the card initially at location to location . To calculate this, let be the matrix obtained from by deleting row and column , and compute the fundamental matrix . Then the expected number of shuffles until the target card starting in position first reaches position is the sum of the values in the -th row of (or the -st row if ).

**Solution**

The answer is interesting and perhaps surprising. First: it typically takes a *long* time for the top card to reach the bottom of the deck, over 100 shuffles on average. Second: this average is actually *exactly* 104 shuffles! More generally, given a deck with cards, the expected number of GSR shuffles to move the top card to the bottom of the deck appears to be exactly . This suggests once again that there may be a nice intuitive argument for this result that I don’t yet see.

Finally, and perhaps most surprising: it takes about that same hundred-or-so shuffles on average to move *any* card to the bottom of the deck!

In the above figure, the *x*-axis indicates the starting position of the card that we want to track. The *y*-axis indicates the expected number of shuffles needed to move that card to the bottom of the deck. For example, even the next-to-bottom card of the deck takes about 102.6 shuffles on average to reach the bottom of the deck.

There are at least a couple of different possible problems here, depending on what constitutes a matching pair of socks. Arguably the most natural setup is that all pairs are distinct (e.g., each pair of my dress socks is a different color), so that each individual sock has exactly one mate. This is what has been described as *the* sock matching problem in the literature; see the references below.

My athletic socks, on the other hand, are essentially identical pairs, with each individual sock being distinguished only by a “left” or “right” label stitched into it, so that each sock may be matched with any of the other “differently-handed” socks. In this case, it’s a nice problem to show that

and thus

But what I found most interesting about this problem is that appears to be very well approximated by , with an error that I conjecture is always less than 1/2, and approaches zero in the limit as grows large. I don’t see how to prove this, though.

**References:**

Whether being on a streak yourself, or trying to defend a player on a streak, on the court this certainly *feels* like a real phenomenon. But is the hot hand a real effect, or just another example of our human tendency to see patterns in randomness?

A famous 1985 paper (Gilovich, Vallone, and Tversky, reference below) argued the latter, analyzing the proportion of successful shots immediately following a streak of three made shots in various settings (NBA field goals, free throws, and a controlled experiment with college players). Not finding any significant increase in proportion of “streak-extending” shots made, the apparent conclusion would be that a past streak has no effect on current success.

But that’s where this puzzle comes in: even if basketball shots are truly *iid* with success probability , we should *expect* a negative bias in the proportion of shots made following a streak, at least compared to the intuitively expected proportion . Miller and Sanjurjo argue that the *absence* of this bias in the 1985 data suggests that the hot hand is *not* just “a cognitive illusion.”

Both papers are interesting reads. In presenting the problem here as a gambling wager, I simplified things somewhat down to a “win, lose, or push” outcome (i.e., were there more streak-extending successes than failures, or fewer), since the resulting exact expected return can be computed more efficiently than the expected *proportion* of successes following a streak:

Given remaining trials (basketball shots, coin flips, whatever) with success probability , noting outcomes following streaks of length , and winning (losing) the overall wager if the number of streak-extending successes is greater (less) than the number of streak-ending failures, the expected return is , computed recursively via

where, using the setup in the previous post where we flip fair coins with probability of heads, looking for heads extending streaks of length , the expected return is about -0.150825.

**References:**

- Gilovich, T., Vallone, R., and Tversky, A., The Hot Hand in Basketball: On the Misperception of Random Sequences,
*Cognitive Psychology*,**17**(3) July 1985, p. 295-314 [PDF] - Miller, Joshua B. and Sanjurjo, Adam, Surprised by the Hot Hand Fallacy? A Truth in the Law of Small Numbers,
*Econometrica*,**86**(6) November 2018, p. 2019-2047 [arXiv]

“If after tossing four heads in a row, the next coin toss also came up heads, it would complete a run of five successive heads. Since the probability of a run of five successive heads is [only] 1/32, a person might believe that the next flip would be more likely to come up tails rather than heads again. This is incorrect and is an example of the gambler’s fallacy.”

That is, having observed a streak of four heads in a row, we are actually just as likely to observe heads *again* on the subsequent fifth flip as we are to observe tails. Similarly, even after betting on red at the roulette wheel and losing four times in a row, we should still expect to win a fifth such bet on red the same stubborn 18/38 of the time (assuming a typical double-zero American wheel).

Right?

So, here is what I think is an interesting puzzle: let’s play a game. I will flip a fair coin times, and prior to each flip, if you have observed a current streak of or more consecutive heads, then make a note of the outcome of the subsequent flip. After all 100 coin flips, tally the noted “streak-following” flips: if there are more heads than tails, then I pay you one dollar. If there are more tails than heads, then you pay me one dollar. (If there are an equal number of heads and tails, then we push.)

If the gambler’s fallacy is indeed a fallacy, then shouldn’t this be a fair bet, i.e., with net zero expected return? But I claim that I have a significant advantage in this game, taking more than 15 cents from you on average every time we play! Following a streak of heads, we expect to observe a much larger proportion of “trend-correcting” tails than “streak-extending” heads.

And there is nothing special or tricky about this particular setup. Try this experiment with a different number of coin flips, or a longer or shorter “target” streak length , or even a roulette-like bias on the coin flip probability. Or instead of focusing only on streaks of consecutive *heads* (i.e., ignoring streaks of tails), look for streaks of *either* or more heads or or more tails, and note whether the subsequent flip is *different*. The effect persists: on average, we observe a larger-than-expected proportion of outcomes that *tend to end the streak*.

We often need “random” numbers to simulate effects that are *practically non-deterministic*, such as measurement noise, reaction times of human operators, etc. However, we also need to be able to reproduce experimental results– for example, if we run hundreds of Monte Carlo iterations of a simulation, and something weird happens in iteration #137, whether a bug or an interesting new behavior, we need to be able to go back and reproduce that specific weirdness, exactly and repeatably.

Pseudo-random number generators are designed to meet these two seemingly conflicting requirements: generate one or more sequences of numbers having the *statistically representative appearance* of randomness, but with a mechanism for exactly reproducing any particular such sequence repeatably.

The motivation for this post is to describe the behavior of pseudo-random number generation in several common modeling and simulation environments: Python, MATLAB, and C++. In particular, given a sequence of random numbers generated in *one* of these environments, how difficult is it to reproduce that same sequence in *another* environment?

**Mersenne Twister**

In many programming languages, including Python, MATLAB, and C++, the default, available-out-of-the-box random number generator is the Mersenne Twister. Unfortunately, despite its ubiquity, this generator is not always implemented in exactly the same way, which can make reproducing simulated randomness across languages tricky. Let’s look at each of these three languages in turn.

(As an aside, it’s worth noting that the Mersenne Twister has an interesting reputation. Although it’s certainly not optimal, it’s also not fundamentally broken, but lately it seems to be fashionable to sniff at it like it is, in favor of more recently developed alternatives. Those alternatives are sometimes justified by qualitative arguments or even quantitative metrics that are, in my opinion, often of little *practical* consequence in many actual applications. But that is another much longer post.)

**Python**

Python is arguably the easiest environment in which to reproduce the behavior of the original reference C implementation of the Mersenne Twister. Python’s built-in `random.random()`

, as well as Numpy’s legacy `numpy.random.random()`

, both use the same reference `genrand_res53`

method for generating a double-precision random number uniformly distributed in the interval [0,1):

- Draw two 32-bit unsigned integer words .
- Concatenate the high 27 bits of with the high 26 bits of .
- Divide the resulting 53-bit integer by , yielding a value in the range .

So given the same generator state, both the built-in and Numpy implementations yield the same sequence of double-precision values as the reference implementation.

Seeding is slightly messier, though. The reference C implementation provides two methods for seeding: `init_genrand`

takes a single 32-bit word as a seed, while `init_by_array`

takes an array of words of arbitrary length. Numpy’s `numpy.random.seed(s)`

behaves like `init_genrand`

when the provided seed is a single 32-bit word. However, the built-in `random.seed(s)`

*always* uses the more general `init_by_array`

, even when the provided seed is just a single word… which yields a different generator state than the corresponding call to `init_genrand`

.

**MATLAB**

Although MATLAB provides several different generators, the Mersenne Twister has been the default since 2005. Its seeding method `rng(s)`

is “almost” exactly like the reference `init_genrand(s)`

— and thus like `numpy.random.seed(s)`

— accepting a single 32-bit word as a seed… except that `s=0`

is special, being shorthand for the “default” seed value `s=5489`

.

The `rand()`

function is also “almost” identical to the reference `genrand_res53`

described above… except that it returns random values in the *open* interval (0,1) instead of the half-open interval [0,1). There is an explicit rejection-and-resample if zero is generated. It’s an interesting question whether this has ever been observed in the wild: it’s confirmed *not* to occur anywhere in the first random draws from any of the sequences indexed by the possible single-word seeds.

**C++**

Both Visual Studio and GCC implementations of the Mersenne Twister in `std::mt19937`

(which is also the `std::default_random_engine`

in both cases) allow for the same single-word seeding as the reference `init_genrand`

(and `numpy.random.seed`

, and MATLAB’s `rng`

excepting the special zero seed behavior mentioned above).

However, the C++ approach to generating random values from `std::uniform_real_distribution<double>`

is different from the reference implementation used by Python and MATLAB:

- Draw two 32-bit unsigned integer words .
- Concatenate all 32 bits of with all 32 bits of .
- Divide the resulting 64-bit integer by .

By using all 64 bits, this has the effect of allowing generation of more than just equally-separated possible values, retaining more bits of precision for values in the interval (0,1/2) where some leading bits are zero.

Another important difference here from the reference implementation is that the two random words are reversed: the *first* 32-bit word forms the *least* significant 32 bits of the fraction, and the second word forms the most significant bits. I am not sure if there is any *theoretically* justified reason for this difference. One possible *practical* justification, though, might be to prevent confusion when comparing with the reference implementation: even for the same seed, these random double-precision values “look” nothing at all like those from the reference implementation. Without the reversal, this sequence would be within the eyeball norm of the reference, matching in the most significant 27 or more bits, but not yielding *exactly* the same sequence.

Some role-playing games involve a “roll and keep” dice mechanic, where you roll some number of dice, but only “keep” a specified number of them with the largest values, where the result of the roll is the sum of the kept dice. For example, rolling five dice and keeping the largest three (sometimes denoted 5d6k3) could yield a score between 3 and 18, inclusive. What is the probability of each possible score?

The motivation for this post is (1) to capture my notes on the general solution to this problem, and (2) to describe some potential optimizations in implementation to handle very large instances of the problem, such as this Hacker Rank version of the problem, where we may be rolling as many as 10,000 dice.

**The solution**

To start, let’s simplify the problem by just rolling and keeping all of the dice without discarding any, since the order statistics are what make this problem complicated. Given dice each with sides, the number of equally likely ways to roll a score is given by

so that the probability of rolling a score is .

If we now consider keeping only of the dice, then we can group the desired outcomes with score by:

- the smallest value among the largest dice that we keep,
- the number of kept dice that are strictly greater than , and
- the number of discarded dice that are strictly less than .

This yields the following summation:

**Implementation details**

At this point, we’re done… except that if the number of dice rolled is large, then a straightforward implementation of this nested summation will be pretty slow, involving calculation of a large number of *very* large multinomial coefficients.

My first thought was to speed up calculation of those very large coefficients via their prime factorization using Legendre’s formula. The paper by Goetgheluck linked at the end of this post describes a more optimized approach, but the following simple Python implementation is already significantly faster than the usual iterative algorithm for sufficiently large inputs (the implementation of `primes(n)`

is left as an exercise for the reader, or a future post):

def binomial(n, k): """Large binomial coefficient n choose k.""" if k < 0 or k > n: return 0 c = 1 for p in primes(n): c = c * p ** (v_fact(n, p) - v_fact(k, p) - v_fact(n - k, p)) return c def v_fact(n, p): """Largest power of prime p dividing n factorial.""" v = 0 while n > 0: n = n // p v = v + n return v

(It’s worth noting that the Hacker Rank problem actually only asks for the number of ways to roll a given total *modulo a prime*, suggesting that a similar but even more effective optimization using Lucas’s theorem is probably intended.)

But we can do much better than this, by observing that in the “usual” iterative algorithm, all of the intermediate products inside the loop are also binomial coefficients… and we need all of them for this problem. That is, we can rewrite the summation as

and then “embed” the usual iterative calculation of the binomial coefficients inside the nested summations. The following implementation eliminates all but a single explicit calculation of a binomial coefficient:

def rolls_keep(s, n, d, m): """Ways to roll sum s with n d-sided dice keeping m largest.""" result = 0 for a in range(1, d + 1): choose_b = 1 for b in range(0, m): sum_c = 0 choose_c = 1 pow_c = 1 for c in range(0, n - m + 1): sum_c = sum_c + choose_c * pow_c choose_c = choose_c * (n - b - c) // (c + 1) pow_c = pow_c * (a - 1) result = result + choose_b * rolls(s - a * m, b, d - a) * sum_c choose_b = choose_b * (n - b) // (b + 1) return result def rolls(s, n, d): """Ways to roll sum s with n d-sided dice.""" if s == 0 and n == 0: return 1 if d == 0: return 0 result = 0 choose_k = 1 for k in range(0, (s - n) // d + 1): result = result + (-1) ** k * choose_k * binomial(s - k * d - 1, n - 1) choose_k = choose_k * (n - k) // (k + 1) return result

**Reference:**

- Goetgheluck, P., Computing Binomial Coefficients,
*The American Mathematical Monthly*,**94**(4) April 1987, p. 360-365 [JSTOR]

In the children’s card game Snap, a deck of cards is shuffled and divided as evenly as possible among two or more players. In alternating turns, each player deals a card from her stack onto a face-up pile in front of her. If at any time, the top cards of any two players’ face-up piles have the same rank, the first player to shout “Snap!” takes both face-up piles and adds them to the bottom of her remaining stack. The objective is to accumulate all of the cards.

It is possible for the players to deal through their entire stacks without a single snap, i.e., at no time do the top cards of two piles have the same rank. Let us call such a game *boring* (a term that may be suggested by, say, your niece to whom you are introducing the game). What is the probability of a boring game of Snap?

One of the reasons I love combinatorics is that it is so easy to ask a hard question. That is, we can pose a series of very similar-sounding problems, all of which are easy to state and understand, but whose minor differences result in major changes in complexity of their solutions. The motivation for this post is to describe a simple children’s card game as an example of this phenomenon.

**Frustration Solitaire**

Before tackling the actual problem described above, let’s consider a slightly modified version of the two-player game that is easier to analyze:

- Instead of dividing a single shuffled deck in half, start with two separate complete shuffled decks, one for each player.
- Instead of alternating turns dealing a card from each player’s stack, at each turn
*both*players*simultaneously*deal a card face-up from their remainder.

Dealing through the entirety of both decks without a snap is effectively equivalent to winning a game of Frustration Solitaire, where a single player deals cards from a single shuffled deck, saying ranks “Ace, two, three, …, king” repeatedly, losing the game if the dealt card ever matches the called rank.

Even within this already-modified context, there are three slight variations that range from very simple to– I think– intractable:

- If a snap requires that both cards match in rank
*and*suit, then we are counting derangements, and the probability of a boring game is approximately , or about 0.367879. - If a snap requires that both cards match in rank only, as in Frustration Solitaire, then we have discussed this problem before here, in the context of a Secret Santa drawing among 13 families each with 4 family members. In this case, the probability of a boring game is approximately 0.0162327.
- If a snap requires that both cards match in rank
*or*in suit… well, although this is still effectively a problem of counting permutations with restricted positions, I*think*this problem is much harder in practice, since the board of restricted positions can’t be nicely decomposed into sub-boards with no rows or columns in common.

**War**

Let’s consider one more modified version of the game, this time actually rolling back one of the changes above: let’s return to playing with a single shuffled deck divided evenly among the two players, but retain the change where the players deal simultaneously from their stacks.

This version of the game has a lot of structure in common with another children’s card game, War. In this case, a boring game of Snap corresponds to War with no war– that is, dealing through both deck halves without any pair of cards matching in rank.

This is a relatively straightforward inclusion-exclusion problem; with a deck with ranks and suits, the probability of a boring game is

where

which for a standard 52-card deck yields a probability of a boring game of about 0.210214.

**Counting “words” with prohibited subwords**

Coming back finally to the original rules of Snap, counting boring games is complicated by the fact that the players *alternate* turning individual cards face-up, rather than simultaneously revealing a *pair* of cards at a time, so that a snap may “start” with either the first or the second player’s deal. Intuitively, consider skipping the initial separation of the deck into halves, and simply deal cards one at a time from the single shuffled deck; a boring game is one in which no two consecutively dealt cards are the same rank.

There is a wonderful paper by Jair Taylor (referenced below) describing a very general but more sophisticated technique for counting arrangements with these types of restrictions. Applying this technique to Snap, the probability of a boring game using a deck with ranks and suits is

where

yielding a probability of approximately 0.0454763, or about once every 22 shuffles, that a game of Snap will be boring.

Once again, however, it’s very easy to make this problem much harder: if we allow a snap to involve adjacent cards matching in rank *or* in suit, what is the resulting probability of a boring game? What if there are more than two players?

**Reference:**

- Taylor, J., Counting words with Laguerre series [arXiv]

What is the median poker hand? More precisely, among all possible five-card poker hands, what hand ranks higher than exactly half of them?

Of course, the solution is possibly not quite unique, since the ranking of hands is a weak order: suits never decide the relative ranking of hands. For example, there are four possible royal flushes, and no one royal flush beats another. So, as a follow-up combinatorics problem: how many “distinct” hand ranks are there? That is, how many equivalence classes are there in the incomparability relation “hand ties with hand ” on the set of possible hands?

]]>x = java.lang.Double(pi); disp(x.isInfinite());

Contrast this with C++, where it is typically necessary to write an additional small amount of “wrapper” C code using the MEX API to bridge the gap between languages.

For the most part, conversions between MATLAB and Java data types are automatic, as expected, and as desired… with one exception: passing a primitive array *by reference*. For example, suppose that we want to call a static method in a third-party Java library that converts an input 6-state vector (as a primitive array of 6 doubles) from one coordinate frame to another, by populating a given “output” array of 6 doubles:

public static int convert(double[] x_in, double[] x_out) { ... }

As far as I can tell, it is impossible to use a method like this directly from MATLAB. The problem is that we can never construct the necessary output buffer `x_out`

, and hold onto it as a reference that could be passed to `convert`

. MATLAB insists on automatically converting any such reference to a Java array of primitive type into its “value” as a MATLAB array of the corresponding native type, leaving the reference to the actual “buffer” behind in the Java address space.

Interestingly, MATLAB’s built-in function javaArray seems like it *might* have been intended for situations like this. But the arrays created by this built-in function are not of the necessary *primitive* data types, but of their “boxed” counterparts, which seems almost entirely useless to me, since I don’t know of any Java libraries whose interface involves *arrays* of *boxed* primitive types.

So, it seems that we need at least *some* additional wrapper Java code to make methods like this usefully accessible from MATLAB. The objective of this post is to provide an implementation that I think fills this gap in a reasonably general way, by using Java’s reflection mechanism to allow creating primitive arrays and passing them by reference to any Java method directly from MATLAB.

(*Aside*: I suppose I get some masochistic enjoyment from repeatedly using the phrase “pass by reference” here, hoping for pedantic complaints arguing that “Java is pass by value, not pass by reference.”)

The code is available on GitHub, as well as the old location here. Following is a simple example showing how it works:

s = java.lang.String('foo'); dst = matlab.JavaArray('java.lang.Character', 3); callJava('getChars', s, int32(0), int32(3), dst, int32(0)); dst.get()

The idea is similar to the `libpointer`

and `calllib`

interface to C shared libraries: a `matlab.JavaArray`

acts like a `libpointer`

, and `callJava`

acts like `calllib`

, automatically “unwrapping” any `matlab.JavaArray`

arguments into their underlying primitive array references.

Finally, a disclaimer: the method dispatch is perhaps overly simple, keying only on the desired method name and provided number of arguments. If your Java class has multiple methods with the same name, and the same number of parameters– regardless of their type, or of the return type– the first method matching these two criteria will be called.

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