Probability of the intersection of two events if they are dependent

Question

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.

Answer 1

You just need to consider the basic event

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

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

Answer 2

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 $P(A\cap B\cap C) = ...?$