s, is n (S)=10 further, it is known that the outcomes favorable to a
certain event E within S, is 6.
Find the odds against event E?

Hi, skeeter1-ga:

Except for the arithmetic typo (4/10 = 0.4 not 0.2) I'd agree with
yaphet-ga.  One really needs an additional assumption, e.g. about
uniform probability distribution, to say anything about the chances of
not E.

Technically skeeter1-ga has asked about "the odds against event E", so
we also need to be a little careful in how the answer is phrased. 
Odds are stated as a ratio of two complementary probabilities. 
Specifically if event E has probability Pr(E), then "the odds against
event E" would be this ratio:

(1 - Pr(E))/Pr(E)

whereas the odds in favor of event E would be the reciprocal of that. 
Here, if we accept Pr(E) = 0.6, then the odds against event E are 2 to
3.  Likewise the odds in favor of E are 3 to 2.

[Odds]
http://www.loyno.edu/~scariano/Math122/Odds.html

regards, mathtalk-ga

Assuming that there is a uniform distribution over S,
Prob(Not E) = 4/10 = 0.2

If we can not make this assumption then we can not know the odds against event E.