An urn contains three green dice and two red dice. Two dice are randomly drawn from the urn in one draw. Give a term with which one can determine the probability that one red die and one green die are drawn.
A green die and a red die are rolled simultaneously. The random variable $X$ describes the sum of the two numbers rolled. Give all values that the random variable $X$ can take and determine the probability $P ( X = 7 )$.
In a municipality there are 6250 households, of which 2250 have a fast internet connection. Two thirds of the households that have a fast internet connection also have a subscription to a streaming service. $46 \%$ of all households have neither a fast internet connection nor a subscription to a streaming service. Consider the following events:\ $A$ : ``A randomly selected household has a fast internet connection.''\ $B$ : ``A randomly selected household has a subscription to a streaming service.''\ Create a completely filled four-field table for the described situation and check whether the events $A$ and $B$ are stochastically independent.
Determine how many households the company would need to contact at minimum so that with a probability of more than $99 \%$ at least one contacted household that does not yet have a fast internet connection would decide to set one up. Assume that every hundredth contacted household that does not yet have a fast internet connection decides to set one up.
The random variable $Z$, which for a fair coin describes the number of occurrences of ``heads'' in four rolls, also has the expected value 2 and analogously $P ( Z = 2 ) = \frac { 3 } { 8 }$. Calculate the variance of $Z$, compare it with the variance of $Y$, and based on this describe a qualitative difference in the probability distributions of $Z$ and $Y$.