How Odds Are Related to Probability

How Odds Are Related to Probability Many times the odds of an event occurring are posted. For example, one might say that a particular sports team is a 2:1 favorite to win the big game. What many people do not realize is that odds such as these are really just a restatement of the probability of an event. Probability compares the number of successes to the total number of attempts made. The odds in favor of an event compares the number of successes to the number of failures.  In what follows, we will see what this means in greater detail. First, we consider a little notation. Notation for Odds We express our odds as a ratio of one number to another. Typically we read ratio A:B as A to B. Each number of these ratios can be multiplied by the same number. So the odds 1:2 is equivalent to saying 5:10. Probability to Odds Probability can be carefully defined using set theory and a few axioms, but the basic idea is that probability uses a real number between zero and one to measure the likelihood of an event occurring. There are a variety of ways to think about how to compute this number. One way is to think about performing an experiment several times. We count the number of times that the experiment is successful and then divide this number by the total number of trials of the experiment. If we have A successes out of a total of N trials, then the probability of success is A/N. But if we instead consider the number of successes versus the number of failures, we are now calculating the odds in favor of an event. If there were N trials and A successes, then there were N - A B failures. So the odds in favor are A to B. We can also express this as A:B. An Example of Probability to Odds In the past five seasons, crosstown football rivals the Quakers and the Comets have played each other with the Comets winning twice and the Quakers winning three times. On the basis of these outcomes, we can calculate the probability the Quakers win and the odds in favor of their winning. There was a total of three wins out of five, so the probability of winning this year is 3/5 0.6 60%. Expressed in terms of odds, we have that there were three wins for the Quakers and two losses, so the odds in favor of them winning are 3:2. Odds to Probability The calculation can go the other way. We can start with odds for an event and then derive its probability. If we know that the odds in favor of an event are A to B, then this means that there were A successes for A B trials. This means that the probability of the event is A/(A B ). An Example of Odds to Probability A clinical trial reports that a new drug has odds of 5 to 1 in favor of curing a disease. What is the probability that this drug will cure the disease? Here we say that for every five times that the drug cures a patient, there is one time where it does not. This gives a probability of 5/6 that the drug will cure a given patient. Why Use Odds? Probability is nice, and gets the job done, so why do we have an alternate way to express it? Odds can be helpful when we want to compare how much larger one probability is relative to another. An event with a probability 75% has odds of 75 to 25. We can simplify this to 3 to 1. This means that the event is three times more likely to occur than not occur.