连续抛掷一枚均匀的硬币,直到出现正面向上, 紧接着出现反面向上为止的概率是多少？

jiewenji

概率论导论 p143 23题

疑问：
1，答案中k-2是怎么统计的？为什么要讨论k-2
2，k-1为什么会出现在概率公式中？概率公式中的k-1与文字讨论部分的k-1有什么关系？是计数么？我看不出这里有排列组合关系。


问题：
现在假定连续抛掷一枚均匀的硬币,直到出现正面向上, 紧接着出现反面向上为止.写出抛掷次数的分布列、期望值和方差.

答案：
If k > 2, there are k -1 sequences that lead to the event { X = k } . One such sequence is H · · · HT, where k -1 heads are followed by a tail. The other k-2 possible sequences are of the form T · · · TH · · · HT, for various lengths of the initial T · · · T segment. For the case where k = 2, there is only one (hence k-1) possible sequence that leads to the event { X = k } , namely the sequence HT. Therefore, for any k ≥ 2,
\(P(X=k)=(k-1)(1/2)^{k }\)