中值概率

我们可以首先确定三个变量的中位数如何大于 3。有两种可能的方法：

1. 事件 $A$：所有三个变量的值都大于 3。
2. 事件$B$：两个变量大于3，一个小于3。

定义了这两个事件后，我们现在的目标是找到： $$P(\text{Median} > 3) = P(A) + P(B)$$ 为了使变量的值大于 3，该值必须落在 3 和 4 之间，对于均匀分布，其概率为 1/4。 $$P(A) = (1/4) * (1/4) * (1/4) = 1/64$$ 对于均匀分布，变量小于 3 的概率就是 1 减去它大于 3 的概率。鉴于这种情况可能发生 3 次，我们乘以 3。 $$P(B) = 3 * ((3/4) * (1/4) * (1/4)) = 9/64$$ 将两者放在一起我们得到： $$P(\text{Median} > 3) = P(A) + P(B) = 1/64 + 9/64 = \boxed{5/32}$$

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Original Explanation

We can start by identifying how the median of the three variables could be greater than 3. There are two possible ways:

1. Event $A$: All three variables have a value greater than 3.
2. Event $B$: Two variables are greater than 3 and one is smaller than 3.

With those two events defined, our goal is to now find:

$$P(\text{Median} > 3) = P(A) + P(B)$$

In order for the variable to have a value greater than 3 the value must fall between 3 and 4, which has a probability of 1/4 for a uniform distribution.

$$P(A) = (1/4) * (1/4) * (1/4) = 1/64$$

For a uniform distribution, the probability that a variable is less than 3, is simply one minus the probability that it is greater than 3. Given this can occur three times we multiply by 3.

$$P(B) = 3 * ((3/4) * (1/4) * (1/4)) = 9/64$$

Putting the two together we get:

$$P(\text{Median} > 3) = P(A) + P(B) = 1/64 + 9/64 = \boxed{5/32}$$