Sometimes, especially when reading news articles, I get the feeling people consider probabilities and odds to be the same thing. For example, here is a Business Insider headline claiming that Nate Silver is predicting “92% odds” of Obama victory. I think what they really mean is that Mr. Silver is predicting a 92% probability of Obama victory. There is a big difference between these two statements! Mathematically, the odds of an event are defined as the probability of the event happening divided by the probability that the event doesn’t happen. So, while a probability can take on any value from 0 to 1 (or, in percentage terms, 0%-100%, the odds can range anywhere from zero to infinity. In fact, when the odds of an event are 1, the probability is only 50%. In the example from above, if the odds of Obama victory were really 92% (=0.92), then the probability of victory would be only 0.48, or 48%. Here are some plots showing the relationship between probability and odds:
Finally, here is another blog post from “Simply Statistics” that illustrates the importance of variance in comparing statistical estimates. The main idea is that if the variance of your estimate is small (ie, that the estimate is very precise), then it could be numerically close to some other comparison value but still be considered “significantly different”.