Negative binomial 当参数 number of success 为1时为 geometric distribution，这两个分布都有两种形式。
一种是trail version, Negative binomial 对应的变量是number of trails until the kth success. Geometric 对应的变量是number of trails until the first success, 对应的mean是 1/p.
另一种是failure version,  Negative binomial 对应的变量是number of failures before the kth success. Geometric 对应的变量是number of failures until the first success, 对应的mean是 (1/p)-1=(1-p)/p. 而这道题对应的是failure version. 把rain 当做success, no rain 当做failure.
那么题中已知信息 the expected number of non-rainy days before the next rain可以转换成 expected number of failures before the next success. 很明显对应的是failure version, mean 为 (1-p)/p.
如果题目问的是 the expected number of days until the next rain可以转换成 expected number of trail until the next success. 才会对应trail version.