Poisson Random Variables - Application problem On a random day, the number of vacant rooms of a big hotel in New York City is 35, on average. What is the probability that next Saturday this hotel has at least 30 vacant rooms? I know the answer is 0.823 but do not understand how they got this.

Define a random variable $X$ to be the number of vacant rooms of a big hotel. Since the title is Poisson Random Variables, let's assume that this random variable is distributed Poisson. Since you know there are 35 vacant rooms on average, you have that $E(X) = 35$, so you know the parameter of your distribution. You are asked to find the probability there are at least 30 vacant rooms, this is just $P(X=30) + P(X = 31) + ....$ or $ 1 - (P(X = 0) + P(X = 1) + ... P(X = 29)) $ Both should evaluate to your desired answer, if I am not mistaken!