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GMAT Practice Question (One of Hundreds)
A six sided die, numbered 1 to 6, is rolled 3 times. If all of the sides have the same surface area, and the die is rolled completely at random, what are the chances of rolling a 5 at least once?
Correct Answer: D
Use the complement of the event in question (i.e., 1 - Probability method).
- We know that the die is rolled three times, and that there is an equal chance that the die will land on any number, because the surface area of all 6 sides is the same.
- We are asked to determine what the probability is that we will get at least one 5 if the die is rolled three times. Therefore, to achieve a roll of 5 at least once, it could be that the die is rolled 5 only once (on the first, second, or third throw), it could be that the die is rolled 5 twice (on any two of the three throws), or it could be that the die is rolled 5 all three times.
- Rather than determining the probability of each of the above scenarios and adding these probabilities together, we can figure out the probability that the number 5 will not be rolled on any of the three rolls and subtract this from 1. Note that there are only two possible scenarios here--either the 5 will be rolled at least once or a 5 will not be rolled at all. Therefore, added together, these two probabilities must equal 1. From this we derive the following formula:
Probability of something happening at least once = 1 minus probability of the thing not happening at all
- So the probability that, on the first throw, a 5 will not be rolled, is the number of actual outcomes where 5 is not thrown (so, 1, 2, 3, 4, or 6 could be thrown, or 5 numbers) divided by the total possible outcomes, 6 (1, 2, 3, 4, 5, and 6). So the probability that a 5 will not be thrown on roll 1 is 5/6
- Similarly, the probability that a 5 will not be thrown on roll 2 is also 5/6, as is the probability that a 5 will not be thrown on roll 3.
- Note that the probability that two events will both happen (or both not happen, as is the case here) is the probability of the first thing happening multiplied by the probability that the second thing will happen.
- Therefore, the probability that a 5 will not be thrown on roll 1 AND roll 2 AND roll 3 is 5/6 * 5/6 * 5/6, which is 25/36 * 5/6, or 125/216.
- We take 1 minus 125/216 to get 91/216, answer choice (D).
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