<s>[INST] Solving $3$x $6$-sided dice of different colours to find the probability of events An experiment consists of $3$ fair, different coloured dice being rolled. The dice are $6$-sided and the sides show numbers $1,\dots,6$. Let $A$ be the event that none of the dice shows numbers $1$ and $2$, and let $B$ be the event that all dice show an odd number.
A) What is the probability of $A$?
B) What is the probability of $B$?
C) What is the probability of $A$ intersecting $B$?
I've solved this question by finding the total number of possible outcomes:
$|S| = 6^3 = 216$
The results were way too long for the marks given which makes me question the method I used for these solutions.
I ended up with:
A) $P(A) = \frac{64}{216} = \frac{8}{27} $ 
B) $P(B) = \frac{26}{216} = \frac{13}{108}$ 
C) $P(A \cap B) = \frac{8}{216} = \frac{1}{27}$
 [/INST] A) The number of ways $3$ dice can not show a $1$ or $2$ is $4\cdot4\cdot4 = 64$. The total number of outcomes is $6\cdot6\cdot6 = 216$
The probability is therefore $\frac{64}{216} = \frac{8}{27}$.
B) The number of ways $3$ dice can show an odd number is $3\cdot3\cdot3 = 27$. Like before, the total number of outcomes is $216$.
The probability is therefore $\frac{27}{216} = \frac{1}{8}$
C) The number of ways $3$ dice cannot be a 1 or 2 or an even number is $2\cdot2\cdot2 = 8$. 
The probability is therefore $\frac{8}{216} = \frac{1}{27}$.
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