Now that we have looked at how to handle probability problems that deal with independent events, we are going to consider how the GMAT can make these problems more complex.

The main way in which the GMAT will do this, is by asking you the **probability of getting “at least” a certain number of a particular outcome** rather than the probability of getting “exactly” that number.

For example, in our last article we considered how to handle a question that asks for the probability of getting *exactly* one head when flipping a coin twice. A more complex problem would ask us the **probability of getting at least two heads when flipping a coin five times.** Now, instead of having one desired outcome – exactly one head – we have four outcomes that are considered desired – two, three, four or five heads. To put that in probability speak, we want two heads OR three heads OR four heads OR five heads. As we remember, all of those “ors” become addition.

This is where an important strategy for advanced probability comes into play. In any situation, the sum of the probabilities of all possible events will add up to one. This means that **on some GMAT problems, such as the one described above, it will be quicker to find the probability of an event NOT happening and subtracting the result from one, than it would be to find the answer directly.** (You can also think of this as “1 minus the probability of something NOT occuring = probability that it WILL occur”.)

The outcomes that do NOT give us one at “least two heads” are: zero heads and one head. We next need to find the probability of each of these events occurring, add those numbers together and subtract from one. But to find those probabilities, we need to veer into the realm of combinations, which will be the topic of the next article in this series. So hold that thought for this question—and stay tuned for our next article!

Bret has been teaching for Kaplan since 2005, and has helped over 1000 students with their GMAT preparation. He spent three years teaching in Manhattan, where he served as an Elite Teacher and a full-time instructor, before moving to London, where he is now the GMAT Master Teacher for Kaplan’s London Center. As the GMAT Master Teacher, Bret trains, observes and mentors teachers, in addition to continuing his own teaching and tutoring, and has taught courses across Europe, including Italy, Ireland, and Germany. Bret contributes to Kaplan’s GMAT curriculum on an on-going basis, and was also a contributor to Kaplan's 2010 GMAT course.