Alice enlists the help of some friends to generate a random number,
using a single unfair coin. She and her friends sit around a table and,
starting with Alice, they take it in turns to toss the coin. Everyone
keeps a count of how many heads they obtain individually. The process
ends as soon as Alice obtains a Head. At this point, Alice multiplies
all her friends' Head counts together to obtain her random number.
As an illustration, suppose Alice is assisted by Bob, Charlie, and
Dawn, who are seated round the table in that order, and that they obtain
the sequence of Head/Tail outcomes THHH—TTTT—THHT—H
beginning and ending with Alice. Then Bob and Charlie each obtain heads, and Dawn obtains head. Alice's random number is
therefore .
Define to be the
expected value of Alice's random number, where is the number of friends helping
(excluding Alice herself), and is
the probability of the coin coming up Tails.
It turns out that, for any fixed , is always a polynomial in . For example, .
Define to be the
coefficient of in the
polynomial . So , , and .
You are given that .
Find .
解决方案
不难直接写出关于 的定义式:
其中,枚举变量 表示 Alice 已经连续抛出了 次反面,在第 次抛出了正面,游戏结束。而 个人分别是在做独立的伯努利实验,因此他们每一个人抛出正面的次数期望为。