I have three records. the records means intervals,
$A: [1, 5]$
$B: [2, 6]$
$C: [4, 6]$
A, B and C are three humidity sensors. Value of A is between 1 to 5, B is between 2 to 6, and C is between 4 to 6
Assume the humidity data is uniform distributed(uniform distribution). I'd like to know which sensor would give the minimal value(lowest humidity), with probability.
Clearly the minimum falls in the ranges $[1,2)$ or $[2,4)$ or $[4,5]$ so the probabilities A, B or C give the minimum respectively are
$$\int_{a=1}^2 \frac{da}{5-1}\cdot1\cdot1 + \int_{a=2}^4 \frac{da}{5-1} \cdot \frac{6-a}{6-2} \cdot 1 + \int_{a=4}^5 \frac{da}{5-1}\cdot \frac{6-a}{6-2}\cdot\frac{6-a}{6-4}$$
$$0 + \int_{b=2}^4 \frac{5-b}{5-1}\cdot \frac{db}{6-2}\cdot 1 + \int_{b=4}^5\cdot \frac{5-b}{5-1} \frac{db}{6-2}\cdot\frac{6-b}{6-4}$$
$$0 + 0 + \int_{c=4}^5 \frac{5-c}{5-1}\cdot \frac{6-c}{6-2}\cdot\frac{dc}{6-4}$$
and if you calculate these they should add up to $1$.