Confusion on basic probability example question

probability theory

Suppose we have the following information:

  1. There is a 60 percent chance that it will rain today

  2. there is a 50 percent chance that it will rain tomorrow.

  3. there is a 30 percent chance that it will not rain either day.

what is the probability that it will rain today or tomorrow?

Now, I don't want to retype the solution since the solution is pretty self-explanatory.

But the solution didn't really mention what is the sample space here, I was thinking the sample space is $S=\{A, B,C\}$ where A represents it will rain today and B: it will rain tomorrow. C: not raining on either day.

But if S is the sample space I can't really make sense between A and B, for example, I would think $A\cap B=\emptyset $, since raining today and tomorrow doesn't necessarily have any connection, then $P(A\cup B)=.6+.5$ which doesn't make any sense.

Best Answer

The reason that the solution didn't mention the sample space here is that the question is a trick question.

In the present problem, you have exactly two simple events:

  • $E_1 ~: ~$ It rains today.
  • $E_2 ~: ~$ It rains tomorrow.

The probability of either Event $E_1$ occurring or Event $E_2$ occurring is irrelevant to solving the problem. Further, consideration of the sample space is (arguably) irrelevant to solving the problem.

Let $F$ denote the event that it does not rain today or tomorrow.

In effect, you are given that $p(F) = 0.3,$ and then asked to compute (in effect) $1 - p(F).$

The whole point of the problem is that the nature of the problem's $3$rd premise, coupled with what you are being asked to compute, renders premises 1 and 2 irrelevant.

However, in answer to your question, as the existing comments have already indicated, anytime that you are given exactly $n$ ($\color{red}{\text{simple, rather than compound}}$) events to consider (i.e. events $E_1, E_2, \cdots, E_n$), your sample space always has exactly $2^n$ elements. That is, each of events $E_k$ either occurs or it doesn't.

The third event discussed in the problem, that of events $E_1$ and $E_2$ both failing to occur is a compound event. That is, it is a combination of the two simple events.

Therefore, when forming the sample space, you should ignore the compound event, and enumerate your sample space as having $2^2$ elements, based on the simple events $E_1, E_2$.

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