Odds vs. Probabilities (Confusion) In logistic regression, I know that Odds and Probabilities are two different things, but still I can't distinguish between them. 
I will be very grateful for your clarifications.
 A: A probability lies between 0 and 1; I presume you're familiar enough with it that we don't need to define probability, but please clarify if you do require a basic definition. [So if I say something like "The probability of at least two heads in three tosses of a coin is 1/2" or "the probability that I miss the bus tomorrow is 1/10" I presume the sense of what that means is already clear.]
Odds are the ratio of a probability (p) to its complement (1-p) ($\text{odds}=\frac{p}{1-p}$. In this case the odds for missing the bus would be $\frac{1/10}{9/10}=1/9$.
[In elementary situations where probability is "number of cases where the event holds"/"total cases", odds are "number of cases where the event holds"/"number of cases where it doesn't".]
[In gambling, odds are usually expression not as a fraction but as a literal ratio ("1:9 for" rather than $\frac{_1}{^9}$), and most usually in terms of odds against (the reciprocal of the ratio, "9:1 against"), unless the odds against fall below 1:1, in which case they're again written the first way around, as odds for and referred to as "odds on"); this keeps the larger number expressed first. If there's no adjective (on, for, against), as in "the odds are 20:1", you would normally assume odds against in gambling situations. You'd flip those around and write as a fraction -- $\frac{1}{20}$ to get odds as a statistician writes them. In some situations you will see odds defined differently to this though.]
To convert odds back to probability, divide the odds by (1+odds). So if the odds are 1/9, the probability is $\frac{1/9}{1+1/9}=\frac{1}{10}$
A: Odds ratio is best understood in the gambling context. If you are throwing a dice, and the winning side is 6, then they say you have 5/1 chance to win, i.e. for every one winning chance there are 5 chances to lose. If the wining sides are 2 and 3, then your odds ratio is 2/1.
Now, you can see how the odds ratio is related to the probabilities: $r=\frac{1-p}{p}$. You can plug probability of losing, instead of winning $\pi=1-p$, then $r=\frac{\pi}{1-\pi}$, basically the same thing.
