Probability questions can be among some of the more advanced and trickier problems you’ll face on the GMAT Quantitative section. Be sure to pay attention to the wording of word problems such as this one; in this case when asked about a scenario with “at least twice”, it will be more efficient to solve for that NOT happening and subtract from 1 (since the probability of something happening plus the probability of that same thing NOT happening should add up to 1, or 100%.)

**Question:**

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

**Solution:**

The key phrase to solving this sample GMAT problem is ‘at least twice.’ This means that out of our five flips, two, three, four and five heads are all desired outcomes. On problems such as this one it is important to remember that we can find the probability that something does NOT happen and subtract that from one, in order to find the probability it DOES happen. In this case, if we flip zero heads or one head, we will NOT have at least two heads. Finding these two probabilities, adding them together (remember, that ‘or’ becomes addition in probability) and subtracting from one will be the fastest way to get to the answer.

Because probability means desired outcomes over possible outcomes, we will need to find each of these for this problem. The first outcome we want 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.

Next, we need to find the number of possible outcomes (our denominator). Each flip has two possible outcomes: heads and tails. In order to find total possible outcomes, we multiply the possible outcomes for each individual flip. As we have five flips, we get 2 x 2 x 2 x 2 x 2 = 32 possible outcomes.

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 one to find the probability it does happen. The math for this is as follows:

1 – 3/16 = 16/16 – 3/16 = 13/16

13/16 is answer choice (D) and is the correct answer.

~Bret Ruber

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