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Conditional probability is the likelihood of an event occurring given that another event has already happened. For example, if 3 of 5 students wear glasses, and 2 of them study math, the probability a glasses-wearing student studies math is 2/3.
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Conditional probability is one of the most interesting as well as practical ideas in mathematics. The essential shift is that the sample space is no longer all possible outcomes but only those that satisfy a given condition. This reduction of the universe of cases changes both the denominator and the way we count favorable outcomes. Questions on the GMAT use this principle to test not only comfort with formulas but also clarity of thought. The challenge lies in interpreting the condition correctly and adjusting the sample space before beginning the calculation. This approach appears in card problems, dice problems, and even in word-based scenarios where a condition is embedded in the language. Developing the habit of narrowing focus to the relevant cases is a vital part of sound GMAT prep, and practicing this habit in exercises as well as mock tests ensures that accuracy and confidence hold strong even under exam pressure.
Conditional probability represents the probability of an event B when it is already known that another event A has occurred.
In these cases, the sample space is not the total set of all possible outcomes. Instead, it is restricted to only those outcomes that meet the condition of the first event.
Question: Three cards are drawn from a pack of 52 cards. What is the probability that all three cards are honor cards, if it is known that all cards drawn are red cards?
Normally, the denominator would be 52C3 because three cards are chosen from 52.
But the condition says that all three cards are red.
Since there are 26 red cards, the denominator becomes 26C3.
Among the 26 red cards, there are 8 red honor cards (4 hearts and 4 diamonds).
The favorable cases are the number of ways of choosing 3 out of these 8 cards, which is 8C3.
Therefore, the required probability = 8C3 ÷ 26C3
Question: Three cards are drawn from a pack of 52 cards. What is the probability that all three cards are aces, if it is known that no card drawn is a picture card?
If the condition is that no card drawn is a picture card, the reasoning changes again.
There are 12 picture cards in the deck (3 in each suit). These are discarded.
The total number of possible cards now becomes 40, so the denominator is 40C3.
If we want honor cards under this restriction, only aces remain, since picture cards have been removed.
There are 4 aces in total. Choosing all 3 cards from these 4 gives 4C3 favorable cases.
Therefore, the required probability = 4C3 ÷ 40C3
Conditional probability teaches us that context shapes the problem. By carefully adjusting the sample space, the calculation becomes precise and meaningful. This discipline is central to mastering probability questions on the GMAT. Building such clarity into your full-length GMAT simulation ensures that even when conditions appear complex, you can calmly reduce them to structured steps and solve with confidence.
Life, like conditional probability, often unfolds within the context of prior events. Our choices and outcomes rarely emerge from an open field; they are shaped by conditions that redefine what is possible. GMAT preparation works the same way. Each practice test, each concept mastered, narrows uncertainty and builds clarity for the next step. In the MBA applications procedure, too, your past experiences frame the way schools view your story, and your mindful actions shape what lies ahead. Learning to recognize conditions, adapt to them, and still move forward with confidence is the essence of true preparation.
