What is the proper way to add odds given a know likelyhood of occurance?
For instance: a coin flip having 50/50 odds of getting heads each time you flip it, what are the odds that you would get heads A LEAST ONCE (not ONLY once) in x number of flips.

Or if there is a 20% chance of rain for each day in a four day stretch, what are the odds that it will rain at least once in that four day stretch?

There’s no general method that will cover all cases, but there are large classes of problems that can be handled similarly.

Your first problem is like that–since each flip of the coin is completely independent of all the other flips before it, the number of heads follows what’s known as a binomial distribution. In this case, the probability of at least one head in n flips is 1 - 1/2[sup]n[/sup].

The second problem is stated so that the weather on any given day doesn’t depend on the weather from any previous days. Under that assumption, the probability of rain at least once is 1 - (4/5)[sup]4[/sup], or 369/625. However, the assumption of independence is very likely not valid, and more complicated models would be needed to address a situation like that.

The easiest way to work this out is to note that if the probability of an event occuring is equal to one minus the probability of the event not occuring. So, the probability of getting no rain for four straight days is 4/5 * 4/5 * 4/5 * 4/5, which is 256/625. The probabilty of getting at least one day of rain is the probability of not getting 4 straight days of no rain, which is 1 - 256/625, which is 369/625, or about 59%.

I think the most important step is distinguishing odds from probability, odds being the one that sounds better.

For example, if you draw a card at random from a standard deck, the odds are 1:3 that it will be a heart, because for every heart in the deck there are three non-hearts. You’re three times as likely to get something else as you are to get a heart.

The probability is 1/4, which means if you do this little experiment a zillion times, it should come up as a heart about 1/4 of the time.

AFAIK, the only way odds are useful in calculation is if they can be converted to probabilities first.