Q:

Lowest score never made

In cricket, a player's score can reach into the hundreds, and the lowest score never made was 229 (as of 2004).  Players have made higher scores, and if anyone made 229 exactly, the new lowest-never-made would be 238.
Assume that a player's score is a Poisson process with parameter $\lambda$.  What proportion of scores will ever be the lowest-never-made?  Or, what is the limiting probability that $N$ is ever the lowest score never made?
This is just idle wondering while cricket takes a break for rain.

A:

Let $X_n$ be the $n$'th score ever recorded (in chronological order).  I'm assuming all $X_n$ are independent Poisson($\lambda$) random variables. 
Let $p(x) = P(X_n = x) = e^{-\lambda} \lambda^x/x!$. For nonnegative integers $s$ let $T(s)$ be the first $n$ for which
$X_n = s$  (with probability $1$ there will be some).  Then $s$ is at some time the lowest score never made if and only if $T(s) > T(t)$ for all positive integers $t < s$.  For any permutation $\pi$ of $\{0,1,\ldots,s\}$, let
$M(\pi)$ be the probability that $T(\pi_0) < T(\pi_1) < \ldots < T(\pi_s)$.
It is easily seen that 
$$M(\pi) = \frac{p(\pi_0)}{p(\pi_0) + \ldots + p(\pi_s)} \frac{p(\pi_1)}{p(\pi_1)+ \ldots + p(\pi_s)} \ldots \frac{p(\pi_{s-1})}{p(\pi_{s-1}) + p(\pi_s)} $$ 
Then the probability $P(A_s)$ that $s$ is at some time the lowest score ever made
is the sum of $M(\pi)$ for all permutations of $\{0,1,\ldots,s\}$ that end
in $s$.  So according to my calculations (with Maple's help)
$$\eqalign{P(A_0) &= M([0]) = 1\cr
P(A_1) &= M([01]) = \dfrac{p(0) }{(p(0)+p(1))} = \dfrac{\lambda}{\lambda+1}\cr
P(A_2) &=  M([012]) + M([102]) = {\frac {8 \left({\lambda}^{2}+\lambda+1\right)}{ \left( {\lambda}^{2}+2\,\lambda+2
 \right)  \left( \lambda+2 \right)  \left( {\lambda}^{2}+2 \right) }}
\cr
P(A_3) &= M([0123]) + M([0213]) + M([1023]) + M([1203]) + M([2013]) + M([2103])\cr
&= {\frac { 648\left( {\lambda}^{5}+3\,{\lambda}^{4}+6\,{\lambda}^{3}+
15\,{\lambda}^{2}+18\,\lambda+18 \right)  \left( {\lambda}^{6}+3\,{
\lambda}^{5}+9\,{\lambda}^{4}+15\,{\lambda}^{3}+12\,{\lambda}^{2}+18\,
\lambda+12 \right) }{ \left( {\lambda}^{3}+3\,{\lambda}^{2}+6\,\lambda
+6 \right)  \left( {\lambda}^{2}+3\,\lambda+6 \right)  \left( \lambda+
3 \right)  \left( {\lambda}^{2}+6 \right)  \left( {\lambda}^{3}+3\,{
\lambda}^{2}+6 \right)  \left( {\lambda}^{3}+6 \right)  \left( {
\lambda}^{3}+6\,\lambda+6 \right) }}
\cr}
$$
etc.

