Probability Using Complement Method on GMAT

Quick Summary

Certain probability problems become simpler when solved through complements. Instead of listing every outcome, calculate the probability of negative cases and subtract from one. Example: At least one head in three fair flips: probability equals 1 − P(all tails) = 1 − (1/2) ³ = 7/8.

Overview

Probability often becomes simpler when approached with logic rather than brute force. Consider a dartboard question where the chance of hitting the bullseye in one throw is one-half and we need to find the number of throws required to bring the probability of hitting the bullseye above 90%. A common mistake is to start listing all possible cases across multiple throws, which grows complicated quickly. A more elegant approach is to calculate the probability of not hitting the bullseye, and then subtract that value from one to find the chance of hitting at least once. This method is not only faster but also easier to scale for larger numbers of attempts. Such clarity of reasoning is crucial for success on the GMAT, where efficiency under time pressure makes all the difference. Building this kind of structured thinking should be part of every serious GMAT preparation course, and practicing it in timed conditions through GMAT mock tests ensures that the skill becomes second nature on test day.

Solving Probability Questions Efficiently Using the Complement Method on the GMAT

Logical probability rewards clarity over computation. Model events correctly, decide when outcomes are independent or mutually exclusive, and use complements to convert at least one into a single calculation. Small wording shifts create distinct cases: exactly one, at least one, at most one, or none. Notice symmetry, boundary cases, and minimal counterexamples to check work without enumeration. With a disciplined checklist, even multi case scenarios compress into decisive steps under time pressure, preserving accuracy while avoiding unnecessary arithmetic and confusion.

An Explanatory Problem

A player has a probability of 1/2 of hitting the bullseye with a single dart. What is the minimum number of darts she must throw to have at least a 90 percent chance of hitting the bullseye at least once?

1. Once
2. Twice
3. Thrice
4. Four times
5. More than four times

Step 1: Probability of Not Hitting in One Throw

The chance of hitting in a single throw is 1/2.

Therefore, the probability of not hitting is also 1/2.

Step 2: Using Complements for Two Throws

Instead of calculating many cases such as hitting once, hitting both times, or missing once, the simpler way is to calculate the chance of not hitting in both throws.

That probability is (1/2 × 1/2) = 1/4.

Hence, the chance of hitting at least once in two throws is 1 − 1/4 = 3/4 or 75 percent.

Step 3: Extending to Three Throws

The chance of missing all three throws is (1/2 × 1/2 × 1/2) = 1/8.

Therefore, the chance of hitting at least once is 1 − 1/8 = 7/8 or 87.5 percent.

Step 4: Extending to Four Throws

The chance of missing all four throws is (1/2 × 1/2 × 1/2 × 1/2) = 1/16.

Hence, the chance of hitting at least once is 1 − 1/16 = 15/16 or 93.75 percent, which is above 90 percent.

The Answer

The player must throw the dart four times to ensure a probability greater than 90 percent of hitting the bullseye at least once.

The Larger Lesson

This example highlights how the complement method offers a quicker and cleaner solution than listing multiple cases. The probability of “at least one success” can always be found by subtracting the probability of “no success” from one. Internalizing this principle sharpens accuracy and efficiency, especially in timed settings. Practicing such approaches regularly through GMAT exercises and GMAT mock builds familiarity with logical shortcuts, strengthens decision making under pressure, and ensures that you approach complex probability problems with clarity rather than unnecessary calculation.

Parting Thought

Logical probability shows that clarity often emerges when we shift perspective. Instead of drowning in countless cases, we step back, simplify, and see the problem from a higher vantage. The GMAT, too, rewards such perspective, valuing reasoning over mechanical effort. The MBA application process follows the same truth: success lies not in trying to cover every detail exhaustively, but in highlighting the essentials with structure and purpose. In life as well, uncertainty is constant, but those who approach it calmly and logic