# Probability for uniform random variables

There are $3$ new drink vending machines and $3$ new snack vending machines installed in a company. Suppose the lifetime (in years) of a drink vending machine is a $[0,5]$-uniform random variable, and the lifetime (in years) of a snack vending machine is a $[0,8]$-uniform random variable. The lifetimes of all the vending machines are independent. Find out the probability that after $4$ years there are exactly two vending machines still in work.

My first question is, are the variables continuous or discrete? At first I thought maybe it's discrete because it's in years, but couldn't a machine be working for, say, 2.5, or 1.75 years, therefore making it continuous? And if it is continuous, how would I find $P(X=4)$ assuming $x$ is the lifetime of the vending machine?

I've also found the PDFs and CDFs of the vending machines but I'm not sure where to go from there.

## 1 Answer

The lifetimes are continuous random variables. A Uniform $[0,5]$ can take on any real value from $0$ to $5$. Like you say, the lifetime can be 1.75 years.

You are not looking for the probability $X=4.$ That would be the probability that a particular vending machine has a lifetime of exactly four years. You are looking for the probability that exactly two of the vending machines have a lifetime of greater than four years (and the other four have a lifetime of less that four years.) Since they are continuous random variables, the probability that any machine's lifetime is exactly four years is zero.

The drink vending machines have lifetimes ranging from $0$ to $5$ years, uniformly distributed, so that means the probability that a drink vending machine has a lifetime of greater than four years is $1/5$ and the probability it dies in less than four years is $4/5.$ Likewise since the snack machines' lifetimes range from $0$ to $8$ years, each has a $4/8=1/2$ probability of living more than four years and a $1/2$ probability of living for less than four years.

So you have three snack machines and three drink machines. For each vending machine, you know the probability it lives more than four years, and you're asked for the probability that exactly two of them do. It's now just a 'coinflip'-type problem and you need to work out the various combinations and their probabilities and add them up.