Question: so I'm taking an intro to probability course and this question was given as part of the first assignment. I've read over notes and course material but nothing seems to be relevant to question such as this (i.e. all outcomes a1, a2, etc. not having equal probability). I've guessed that I need to incorporate partitions in some way, but honestly I'm at a loss.

I was wondering if somebody could point me in the right direction. Given this is for an assignment, I'm not looking for a solution, just a little bit of advice. The question is as follows:

Let the sample space Ω = {a1, a2, a3, a4, a5, a6} be given. Assume that: Let A = {a1, a2, a3}, B = {a2, a4}, C =
{a4, a5, a6}, D = {a2} and E = {a4}. If P(A) = P(B) = P(C) and P(D) = 2P(E). Find:

(a) P(A ∩ D')

(b) P(A ∪ E)

(c) P(A|B')

Thanks in advance!

Answer: 
> 
> Let the sample space $Ω = \{a\_1, a\_2, a\_3, a\_4, a\_5, a\_6\}$ be given. Assume that: Let $A = \{a\_1, a\_2, a3\}, B =\{a\_2, a\_4\}, C = \{a\_4, a\_5, a\_6\}, D = \{a\_2\}$ and $E = \{a\_4\}$. If $P(A) = P(B) = P(C)$ and $P(D) = 2P(E)$.
> 
> 
> 

$$\begin{array}{|l|l:6l|}\hline & a\_1 & a\_2 & a\_3 & a\_4 & a\_5 & a\_6 \\ \hline A & \checkmark & 2x & \checkmark \\ \hdashline B && 2x && x \\ \hdashline C &&&& x & \checkmark & \checkmark \\ \hdashline D && 2x \\ \hdashline E &&&& x \\ \hline \end{array}$$

$P(E)=x, P(D) = 2x, P(A) = P(B) = P(C) =3x$

Also $P(A)+P(C)=1$ so...
Work it out where the probability of a1, a2... a6 is equal. For a, that's the probability for a1 or a3 but not a2, which is P(a1, a3), which is 1/3. For b, which is for A or E, which is P(a1, a2, a3, a4), which is 2/3. I expect you can work out the third one. (If you can't, ask me.)