Vamos a generalizar los problemas más.
El juego de cartas: dibujo sin reemplazo
Observe que de acuerdo a la descripción de la tarjeta de juego, cada conjunto de tres empates es sin reemplazo. Generalizar "tres" por $k\ge 1$ y supongamos que hay $N$ distintos valores de las cartas.
$A$ dibuja $k$ de los valores (sin reemplazo) $a_1 \lt a_2 \lt \cdots \lt a_k$ a partir de una población de números de $x_1 \lt x_2 \lt \cdots \lt x_N$ donde $x_i$ probabilidad de $1/N$ de ser dibujado. Después de la colocación de los valores, $B$ dibuja $k$ valores $b_1 \lt b_2 \lt \cdots \lt b_k$. ¿Cuál es la probabilidad de que $a_1 \gt b_k$?
Podemos desglosar esta información por el valor de $b_k$, que puede variar de $x_k$ través $x_{N-k}$. El evento $a_1 \gt b_k = x_m$ significa que todos los $k-1$ $B$'s valores restantes proceden del conjunto de $x_1, \ldots, x_{m-1}$ mientras que los $k$ $A$'s valores fueron tomados de conjunto $x_{m+1}, x_{m+2}, \ldots, x_N$. La primera se puede hacer en $\binom{m-1}{k-1}$ formas mientras que la segunda puede ser hecho en $\binom{N-m}{k}$ maneras. El número total de resultados equiprobables es $\binom{N}{k}\binom{N}{k}$, independientemente. Por lo tanto la respuesta es
$$\binom{N}{k}^{-2} \sum_{m=k}^{N-k} \binom{m-1}{k-1}\binom{N-m}{k} = \binom{N}{k}^{-2}\binom{N-k}{k}\frac{N!}{2^{k} (2 k-1)\text{!!} (N-k)!}.$$
For the cards, $N=32$ and $k=3$, giving $1827/49600\aprox 0.0368347$.
The die game: drawing with replacement
When, as in the dice problem, the draws are with replacement, the same idea applies but the probabilities differ a little. We can decompose the event "maximum of $B$ is less than the minimum of $A$" by the value attained by $\max(B)$, for instance. Let that value be $x_m$. When $G$ is the distribution law of $B$, the chance that all values in $B$ do not exceed $x_m$, but at least one equals it, is
$$G(x_m)^k - G(x_{m-1})^k.$$
Let $F$ be the distribution law of $A$. The chance that all values in $A$ exceed $x_m$ is
$$(1 - F(x_m))^k.$$
Consequently, since $1\le m\lt N$, the solution in general is
$$\sum_{m=1}^{N-1} (G(x_m)^k - G(x_{m-1})^k)(1 - F(x_m))^k.$$
In the present setting $G(x_m) = F(x_m) = m/N$ because all values have equal probabilities. For the dice, $N=6$ and $k=3$, giving $481/15552 \aprox 0.0309285$ for the answer.
If we were to draw the cards with replacement, $N=32$, giving $\frac{3056755}{67108864} \approx 0.0455492$.
The limiting game: both players draw without any replacement
This is the easiest of the three: of the $\binom{2k}{k}$ different and equiprobable configurations of the relative orders of $$ and $B$, in only one do all elements of $B$ precede those of $Un$. The chance of this therefore equals
$$\frac{1}{\binom{2k}{k}} = \frac{(k!)^2}{(2k!)}.$$
When $k=3$, this is $1/20 = 0.05$. Its closeness to the preceding answers--$0.037$ and $0.031$--gives us some assurance those answers are in the right ballpark. The simplicity of this formula also lets us approximate the answer when $n$ is large.
Mathematica computed the exact solutions quoted here. The following R code also computes them (in double precision rather than exactly) and runs simulations to evaluate the correctness of the analytical solutions. These simulations conduct repeated z-tests of the solutions. The mean of the z-values should be zero, but will differ slightly due to random variation. The distribution of the z-values should be approximately Standard Normal (with a little allowable deviation due to the small finite values of $n$ and $k$). Aquí están (por 999 iteraciones cada una), lo que demuestra la exactitud de estas respuestas:
![Figure]()
#
# Simulate a solution and, optionally, test it against an expected value.
#
f <- function(n, k, n.iter=1e3, replace=TRUE, expectation) {
sim <- replicate(n.iter, {
max(sample(1:n, k, replace=replace)) < min(sample(1:n, k, replace=replace))
})
p <- mean(sim)
if (missing(expectation)) return(p)
se <- sqrt(p*(1-p)/n.iter)
return ((p - expectation) / se)
}
#
# Compute the analytical answers.
#
expectation <- function(n, k) {
z <- lgamma(n+1) + lchoose(n-k, k) + 0.5*log(pi)
z <- z - 2*k*log(2) - lgamma(n-k+1) - lgamma(k+1/2)
y <- 2 * lchoose(n, k)
exp(z - y)
}
expectation.replace <- function(n, k) {
g <- (0:n / n)^k
f <- rev(g)[-1] # Chance the min of `k` values exceeds `m`
g <- g[-1] - g[-(n+1)] # Chance the max of `k` values is `m`
sum(f * g)
}
#
# Show simulated and analytical answers.
#
c(expectation=expectation.replace(6, 3), simulated=f(6, 3, 1e5))
c(expectation=expectation(32, 3), simulated=f(32, 3, 1e5, FALSE))
#
# Run repeated tests of the two games.
# If the z-values approximate a standard Normal distribution,
# we can be confident the given expectation is correct.
#
par(mfrow=c(1,2))
z.values <- replicate(999, f(6, 3, replace=TRUE, expectation=expectation.replace(6, 3)))
hist(z.values, main="Dice (with replacement)")
abline(v=mean(z.values), col="Red", lwd=2)
z.values <- replicate(999, f(32, 3, replace=FALSE, expectation=expectation(32, 3)))
hist(z.values, main="Cards (without replacement)")
abline(v=mean(z.values), col="Red", lwd=2)
