I am having this problem: The rainfall in April in Famagusta follows a uniform distribution between $0.5$ inches and $3.0$ inches. I am confused in finding the probability that the amount of rainfall will be exactly 1 inch of rain for the month.
Based on the formula of a uniform probability distribution we have: $P(X=x)=\frac1{2.5}$ for every $x$ on the closed interval $[0.5; 3]$ and consequently $P(X=1)=\frac1{2.5}=0.4$.
On the other hand, based on the rectangular shape of the uniform probability distribution we have: $P(X=1)=$(height)(base)$=(\frac1{2.5}) (1-1)=0.$
But, $0$ is distinct from $0.4$. So what is the probability of exactly $1$ inch of rain? Which of the methods is wrong?
Please give me a hand.
Best Answer
Your problem is here:
That "formula" works when your sample space is finite and you use the number of points in the denominator. But your sample space is an interval. It's a finite interval, but it contains infinitely many points. The probability that you get exactly one inch of rainfall is $0$.