# What is the sample space and events in predicting the rain today and tomorrow?

What is the sample space(S) and events for the following question ?

For instance, if we were to ask someone what he thought the chances were of

(a) rain today,

(b) rain tomorrow,

(c) rain both today and tomorrow,

(d) rain either today or tomorrow

Let $$S$$ be the set of outcomes. There are two properties that $$S$$ must fulfill in order to be considered a sample space:

• The events of $$S$$ are mutually exclusive
• Every outcome of your trial will lie on your sample space

So for $$a$$ we can define two mutually exhaustive events:

• $$R$$ is the event it rains today
• $$R^c$$ is the event it doesn't rain today

Then your sample space will be $$S =\left\{R, R^c \right\}$$

Similarly, for $$c$$:

• $$R_0$$ is the event it rains today
• $$R_0^c$$ is the event it doesn't rain today
• $$R_1$$ is the event it rains tomorrow
• $$R_1^c$$ is the event it doesn't rain tomorrow

The sample space will be $$S=\left\{R_0R_1, R_0R_1^c, R_0^cR_1,R_0^cR1^c\right\}$$

So with that in mind, what is the sample spaces in the cases $$b$$ and $$d$$?