# Probability of the intersection of two events if they are dependent

A system was to select a string uniformly at random from the set {RRRR, RS, STT} and then select a letter uniformly at random from the selected string.

The system is run twice (outputting two letters) Let A be the event that the first output is R. Let B be the event that the second output is S. Let C be the event that the two outputs are the same letter

Q1. How would you find Pr(A∩C)? A and C would be dependent events, as the occurrence of A affects the likelihood of C.

Q2. Once finding Pr(A∩C), is it possible to find Pr(A∩B∩C)?

Thanks for any input/help.

You just need to consider the basic event

$$\mathrm{Pr}(A\cap C)=\mathrm{Pr}(\text{Result is }RR)=\frac{1}{4}$$

$$\mathrm{Pr}(A\cap B\cap C)=\mathrm{Pr}(\text{Result is }RR \text{ and Result is }RS)=0$$

HINT

$$A$$ and $$C$$ are certainly dependent, but that just means you cannot use $$P(A \cap C) = P(A)P(C)$$, which is no problem. Instead just do it from first principles.

$$A \cap C =$$ first letter is $$R$$ and second letter is the same, i.e. also $$R$$. So $$P(A \cap C) = P(R, R) = ...?$$

$$A \cap B \cap C =$$ first $$R$$, second $$S$$, and they are the same. Clearly, this is impossible because $$R \neq S$$ so $$P(A\cap B\cap C) = ...?$$