Probability is one of the trickier subject areas for GMAT students and tutors alike. As hard as it is for GMAT test takers to learn the ins and outs of combinatorics, for example, it is just as hard for tutors to justify precious preparation time focused on such a narrow and complicated subject. On an average GMAT, students will see at most five to seven probability problems, not nearly the number of algebra or geometry problems, to use two other more common subject areas.
As such, students should develop a simple approach to probability problems that allows them to maximize their chances without mastering the content. Having tutored numerous MBA hopefuls, I have devised a three-step process that I will share using the following practice problem:
If Arthur flips a fair, two-sided coin five times, what is the probability that he will land heads exactly three times?
Step One: Find the denominator – the total number of possibilities in the set.
Probability problems essentially ask students to find the number of times a subset occurs within a larger set. The denominator of the answer represents the larger set. In this case, the denominator is the total number of possible sequences of H and T (Heads and Tails) written out five times: HHHHH, HHHHT, HHHTH, etc.
When flipping a coin, we know that there are four possible sequences for two flips – HH, HT, TH, and TT – eight possible sequences for three flips, and so on. In other words, the total number of possibilities equals 2 raised to the number of flips. In this case, 2 raised 5 times equals 32.
Knowing this, we can eliminate B. 5/6 and D. 2/5, since denominators 6 and 5 do not divide evenly into 32. Two down, three to go!
Step Two: Find the numerator – the specified subset – even if you must count it out!
In this example, we must determine how many five letter combinations exist using H & T, in which there are exactly three H’s. An advanced student knows that this is a combinatorics problem that can be solved with the equation 5!/3!*(5-3)! = 10. However, even a novice can solve it without knowing this equation simply by writing out all the possibilities! The student should attempt to be as organized as possible, writing along the lines of:
Notice how T’s are moved to the left one-by-one to avoid double-counting or missing a sequence. Though this task seems daunting, it takes the average GMAT student far less time and proves much more intuitive than most relevant equations.
Step Three: Sanity check your answer with logic.
This is the easiest yet most important step. We are left with 10/32. Does this answer make sense given the parameters of the problem?
First of all, 10/32 is a positive fraction between 0 and 1 – a good start. Many students, deadlocked in calculation-mode rather than in logical reasoning, end up with numbers below 0 or greater than 1 – answers that could not possibly suffice for a probability question.
Secondly, a number close to 1/3 seems logically correct given the parameters. Three heads is one of six possible amounts for the total number of heads in a series of five rolls – 0, 1, 2, 4, and 5 being the other five. That being said, it is tied for most common, with two heads. In this context, 10/32 makes sense, and the student can feel confident choosing A. 5/16, having undertaken a structured and logical – if imperfect – approach to the problem.