Given we know:
$\displaystyle P(e) = .7,\; P(e | \neg s) = .15,\; P(\neg e|s) = .1
$

Find P(s).
$\displaystyle
P(\neg e | s) = P(\neg e \cap s) / P(s) = .1$

$\displaystyle .1 P(s) = P(\neg e \cap s) = P(s \cap \neg e) = P(s) - P(s \cap e)$ <-- is this true?

=> $\displaystyle .9 P(s) = P(s \cap e)$ -- (1)

$\displaystyle P(e|\neg s) = P(e \cap \neg s)/P(\neg s) = .15$

=> $\displaystyle .15(1-P(s)) = P(e) - P(e \cap s)$

since $\displaystyle P(A \cap B) = P(A) - P(A \cap \neg B)\quad \forall A, B.$ <-- not sure about this identity, is it true?

=> $\displaystyle .15 - .15 P(s) = .7 - .9 P(s)$ (by (1))

$\displaystyle .75 P(s) = .55 \Rightarrow P(s) \approx .7333$

is it correct?