# GMAT Problem Dissection: Probability Part 2

July 25, 2013 by

Alright, you’ve been studying the GMAT basics. You understand what is really being tested along with the problem types, methods, and strategies. You’ve even memorized all the basic formulas and taken a dive into some of the more challenging content areas. So what’s next? That’s simple, practice…over and over and over again. If you do this, you are likely to start hitting tough probability questions at some point. Along with combinations and permutations, this is a content area that the GMAT can use to up the degree of difficulty quite a bit. So let’s take a look at a tough probability question and break it down:

A fair coin is tossed five times. What is the probability that it lands heads up at least twice?

(A) 1/16

(B) 5/16

(C) 2/5

(D) 13/16

(E) 27/32

Probability means desired outcomes over possible outcomes. Thus, we will need to find each to solve this problem. Let’s look at the denominator (possible outcomes) first. Each flip has two possible outcomes: heads and tails. Just like in a GMAT permutations question when we are trying to determine the total number of codes possible or 4-digit numbers, we would multiply these individual probabilities together in order to find total possible outcomes. Therefore, there are 2x2x2x2x2 = 2^5 = 32 total possibilities.

Next we look at our numerator (desired outcomes). The first outcome we need to consider is zero heads. Only one way exists for this to happen: all of the flips come up as tails. Thus, we have ONE desired outcome in which zero heads appear. The second outcome we are looking for is exactly one head. Five outcomes would provide one head, as the head could be first, with all other flips coming up tails, the head could be second, with all other flips coming up tails, and so on.

Therefore, in total we have six outcomes that do not give us at least two heads.

TTTTT

HTTTT

THTTT

TTHTT

TTTHT

TTTTH

When we put these together we have a 6/32, or 3/16, probability of not flipping at least two heads. Since we found the probability of what we do not want to happen, we still need to subtract our result from 1 to find the probability it does happen. The math for this is as follows:

1 – 3/16 = 16/16 – 3/16 = 13/16