Statistics 简明教程

Statistics - Geometric Probability Distribution

几何分布是负二项分布的一个特例。它处理单次成功所需的试验次数。因此，几何分布是负二项分布，其中成功的次数 (r) 等于 1。

The geometric distribution is a special case of the negative binomial distribution. It deals with the number of trials required for a single success. Thus, the geometric distribution is a negative binomial distribution where the number of successes (r) is equal to 1.

Formula

其中——

Where −

  1. ${p}$ = probability of success for single trial.

  2. ${q}$ = probability of failure for a single trial (1-p)

  3. ${x}$ = the number of failures before a success.

  4. ${P(X-x)}$ = Probability of x successes in n trials.

Example

Problem Statement:

Problem Statement:

在游乐场，如果一名选手从一定距离将一个环套在钉子上，则他有资格获得奖品。观察发现，只有 30% 的选手能够做到这一点。如果某人获得 5 次机会，他已经错过了 4 次机会，那么他赢得奖品的概率是多少？

In an amusement fair, a competitor is entitled for a prize if he throws a ring on a peg from a certain distance. It is observed that only 30% of the competitors are able to do this. If someone is given 5 chances, what is the probability of his winning the prize when he has already missed 4 chances?

Solution:

Solution:

如果某人已经错过了四次机会，并且必须在第五次机会中获胜，那么这是一个在 5 次试验中取得首次成功的概率实验。问题陈述还表明概率分布是几何分布。成功的概率由几何分布公式给出：

If someone has already missed four chances and has to win in the fifth chance, then it is a probability experiment of getting the first success in 5 trials. The problem statement also suggests the probability distribution to be geometric. The probability of success is given by the geometric distribution formula:

其中——

Where −

  1. ${p = 30 \% = 0.3 }$

  2. ${x = 5}$ = the number of failures before a success.

因此，所需的概率：

Therefore, the required probability: