The Problem of the Points 計點問題
- Example (The Problem of the Points)
- Independent trials, resulting in a success with probability p and a failure
with probability 1-p, are performed. What is the probability that nsuccesses occur before m failures? If we think of A and B as playing a
game such that A gains 1 point when a success occurs and B gains 1 point
when a failure occurs, then the desired probability is the probability that
A would win if the game were to be continued in a position where A needed
n and B needed m more points to win.
- Here, we present two solutions.
Let Pn,m the probability that n successes occur before m failures. By
conditioning on the outcome of the first trial we obtain:
By using the obvious boundary conditions
Pn,0=0, P0,m=1, these
equations can be solved for Pn,m.
For n successes to occur before m failures, it is necessary and sufficient
that there be at least n successes in the first m+n-1 trials. (Even if the
game were to end before a total of m+n-1 trials were completed, we could still
imagine that the necessary additional trials were performed.) This is true, for
if there are at least n successes in the first m+n-1 trials, there could be
at most m-1 failures.
On the other hand, if there were fewer than n successes in the first m+n-1trials, there were fewer than n successes in the first m+n-1 trials, there
would have to be at least m failures in that same number of trials; thus nsuccesses would not occur before m failures.
Hence, as the probability of exactly k successes in m+n-1 trials is,
we see that the desired probability of
n successes before m failures is
where the last identity follows from the substitution i=2n-2-k.Thus
and so the first player is entitled to
As an illustration of the problem of the points suppose that 2 players each put
and each of them has an equal chance of winning each point (p=1/2).
If n points are required to win and the first player has 1 point and the
second none, then the first player is entitled to