Question: Given that events S and T such that $P(S)=\dfrac{1}{5}$, $P(T)=\dfrac{1}{4}$, and $P[(S \cap T') \cup (S' \cap T) ]=\dfrac{1}{6}$.

i) Find $P[(S \cap T)$ and determine if the events S and T are independent. 

ii) Find $P[(S \cap T') / S ]$.

Part (i), I am trying to find the probability of $P(S \cap T)$,

$P[(S \cap T') \cup (S' \cap T) ]=\dfrac{1}{6}$

$P(S \cap T')+P(S' \cap T)-P[(S \cap T') \cap (S' \cap T) ]=\dfrac{1}{6}$

$P(S)-P(S \cap T)+P(T)-P(S \cap T)-P[(S \cap T') \cap (S' \cap T) ]=\dfrac{1}{6}$

$P(S)+P(T)-2P(S \cap T)-P[(S \cap T') \cap (S' \cap T) ]=\dfrac{1}{6}$

$P(S)+P(T)-2P(S \cap T)-P[(S \cap T') \cap (S' \cap T) ]=\dfrac{1}{6}$

$\dfrac{1}{5}+\dfrac{1}{4}-2P(S \cap T)-P[(S \cap T') \cap (S' \cap T) ]=\dfrac{1}{6}$

$P(S \cap T)=\left(\dfrac{17}{60}-P[(S \cap T') \cap (S' \cap T)]\right)\dfrac{1}{2} $

I am not sure how to get the value of $P[(S \cap T') \cap (S' \cap T)]$ and how to check if S and T are independent or not, can someone help me?

For part (ii), what is "/S" inside $P[(S \cap T') / S ]$?

Thank you in advance.

Answer: 
By the Inclusion-Exclusion Principle,
$$\Pr(S \cup T) = \Pr(S) + \Pr(T) - \Pr(S \cap T)$$
Since $S \cup T$ is the union of the disjoint sets $S \cap T'$, $S \cap T$, and $S' \cap T$,
$$\Pr(S \cup T) = \Pr(S \cap T') + \Pr(S \cap T) + \Pr(S' \cap T)$$
Since the sets $S' \cap T$ and $S \cap T'$ are disjoint,
$$\Pr[(S \cap T') \cup (S' \cap T)] = \Pr(S \cap T') + \Pr(S' \cap T)$$
Equating the two results for $\Pr(S \cup T)$ yields
$$\Pr(S) + \Pr(T) - \Pr(S \cap T) = \Pr(S \cap T') + \Pr(S \cap T) + \Pr(S' \cap T)$$
Solving for $\Pr(S \cap T)$ yields
$$\Pr(S \cap T) = \frac{1}{2}[\Pr(S) + \Pr(T) - \Pr(S \cap T)' - \Pr(S' \cap T)]$$

Since you know $\Pr(S)$, $\Pr(T)$, and $\Pr[(S \cap T') \cup (S' \cap T)]$, you can substitute for these values in equation to find $\Pr(S \cap T)$.

Events $S$ and $T$ are independent if $\Pr(S \cap T) = \Pr(S)\Pr(T)$.
Set $(S\cap T^{\complement})\cup(S^{\complement}\cap T)$ is usually denoted as $S\Delta T$.

Observe that it is a union of disjoint sets so that:
$$P(S\Delta T)=P(S\cap T^{\complement})+P(S^{\complement}\cap T)$$

Make a Venn-diagram and set:

* $a:=P(S\cap T^{\complement})$
* $b:=P(S^{\complement}\cap T)$
* $c:=P(S\cap T)$

Then we have the equalities:

* $a+c=\frac15$
* $b+c=\frac14$
* $a+b=\frac16$

This can easily be solved and results in:

* $a=\frac7{120}$
* $b=\frac{13}{120}$
* $c=\frac{17}{120}$

I leave the rest to you.

---

(ii) Probably you are asked to find: $$P(S\cap T^{\complement}\mid S)$$denoting the probability that event $S\cap T^{\complement}$ *under the condition* that event $S$ occurs.

You can use the general rule:$$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$$