# probability greater than the median

When given a set of distinct numbers, the probability of choosing a number greater than the median is 0.5 correct? Is there a condition that the probability would not equal 0.5?

• The probability need not be $0.5$ : If we have the numbers $1,3,5$, the median is $3$. The probability to choose a number greater than $3$ is $\frac{1}{3}$. Your statement gets correct, if we know that we have an even number of numbers. – Peter Nov 30 '18 at 22:49
• @Peter. Or if you have an even number but several are equal to the median... – fleablood Nov 30 '18 at 22:54
• @fleablood Can this happen if we have distinct numbers ? – Peter Nov 30 '18 at 22:56
• If you have distinct numbers, no, and ... oh, I missed that that was a requirment. If there are $2n$ distinct elements the prob is $\frac 12$. and if there are $2n + 1$ distinct elements the prob is $\frac n{2n+1}$. If the elements are not distinct then ... it's anyones game. – fleablood Nov 30 '18 at 23:12

• It's also true for even numbers and some numbers are equal to the median. Say. ${1,5,5,5,7,8}$ Median is $5$. Pro of less is 1/6. Prob of more is 1/3. Prob of exactly is 1/2. – fleablood Nov 30 '18 at 22:57
• @fleablood On the other hand, no value is equal to the median in $\{1,\,2,\,3,\,4\}$. – J.G. Nov 30 '18 at 22:58
• Okay, .... I missed the requirement that the elements are distinct.... so in that case the prob is $\frac {\lfloor \frac n2\rfloor}n$ which is $\frac 12$ iff and only if $n$ is even. You're right. – fleablood Nov 30 '18 at 23:15