Question
There are 20 people in a room. Assume each person’s birthday is equally likely to be any of the 365 days of the year. (a) What is the probability that no one in the room share the same birthday? [5] (b) What is the probability that exactly two people have the same birthday? [5] (c) What is the probability that two or more people have the same birthday?
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Answer to a math question There are 20 people in a room. Assume each person’s birthday is equally likely to be any of the 365 days of the year. (a) What is the probability that no one in the room share the same birthday? [5] (b) What is the probability that exactly two people have the same birthday? [5] (c) What is the probability that two or more people have the same birthday?
Dexter
4.7101 Answers
已知信息：
- 一个房间里有20个人。
- 每个人的生日都有可能出现在一年 365 天中的任何一天。
（a）房间里没有人同天生日的概率：
第一个人的生日可以是 365 天内的任何一天。
对于第二个人，剩余 364 天（共 365 天）可供选择，因此他们的生日与第一个人的生日不同。
对于第三个人，剩余 363 天（共 365 天）可供选择，因此他们的生日与前两个人不同。
继续这个过程，没有人同天生日的概率是：
概率 = (365/365) × (364/365) × (363/365) × ... × (346/365)
概率 = \frac{\left(365\cdot364\cdot363\ldots..346\right)}{365^{20}}
（b）恰好两个人有同一天生日的概率：
步骤1：计算从20个人中选择2个人的方法数。
选择 2 个人的方式数 = 20 的组合取 2 = (20 × 19) / (2 × 1) = 190
第 2 步：计算选择两个人共享生日的特定日子的方法数。
选择特定日期的方式数 = 365
步骤3：计算将剩余18个人分配到另外364天的方法数。
剩余18人的分配方式数=365p18/365^18
步骤 4：将步骤 1、2 和 3 的乘积除以可能结果的总数来计算概率。
可能结果总数 = 365^20
概率 = (190 × 365 × 365p18 / (365^18)) / (365^20)
（c）两个或两个以上的人有相同生日的概率：
两人或两人以上生日相同的概率 = 1 - 无人生日相同的概率
概率 = 1 - (365p20 / (365^20))
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