double checking my answers with Probability for or against 1.Find the odds against an event E when pr(E) = 5/6
2.Find the probability of an event when the odds for the event are 6:4
this is what I got for my answer but I am not sure


*

*5/1

*3/5
 A: There are two common ways of representing the probability of something happening.  The first is as a ratio of how many times you expect a success to happen out of how many times total you try.  A probability like this is the more common definition and for discrete spaces is defined as $Pr(A) = \dfrac{|A|}{|X|}$.  In general, if you see the probability written with a / or as a fraction, this is the definition they usually go with.
In your example, $Pr(A)=\frac{5}{6}$.  The opposite probability (the probability that what it is you are talking about doesn't happen) is given by $Pr(A^c) = 1 - Pr(A)$ and in your example $Pr(A^c)= 1- \frac{5}{6} = \frac{1}{6}$.  Continued below after defining more.

There is another way of expressing probabilities (though less common) and that is as a ratio of how many times a success happens compared to how many times nonsuccesses occur.  In this case, it is commonly written with a colon ':' inbetween the numbers.  That is: $Odds(A) = Pr(A):Pr(A^c) = |A|:|A^c|$  In your example, the odds are $6:4$ for.  In this case, you can calculate the probability of the event by doing $Pr(A) = \dfrac{|A|}{|A|+|A^c|}$.  In your example, $=\dfrac{6}{6+4} = \frac{3}{5}$

Going back to the first example, the question was technically to find the Odds against, not the probability of the opposite, so we do $Odds(A^c) = Pr(A^c):Pr(A) = \frac{1}{6}:\frac{5}{6}$ and since they are ratios, the ratio is the same even if we multiply, so commonly we avoid fractions, it simplifies to $=1:5$
