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Updated on: 01 Jul 2025, 08:23
Originally posted by Sajjad1994 on 01 Jul 2025, 08:23.
Last edited by Sajjad1994 on 01 Jul 2025, 08:23, edited 1 time in total.
Last edited by Sajjad1994 on 01 Jul 2025, 08:23, edited 1 time in total.
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A die is rolled n times, where n is at least 3.
| Quantity A | Quantity B |
| The probability that at least one of the throws yields a 6 | \(\frac{1}{2}\) |
08 Jul 2025, 11:09
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OE
The easiest way to compute the probability in question is through the “1 − x” shortcut. To do so, imagine the opposite of the event of interest, namely, that none of the n throws yields a 6. The probability of a single throw not yielding a 6 is \(\frac{5}{6}\), and because each throw is independent, the cumulative probability of none of the n throws yielding a 6 is found by multiplication:
Powers of fractions less than one decrease as the exponent increases, so this probability will become very small for large values of n, such that the probability of getting at least one 6 (which is \(1-[\frac{5}{6}]\)) will come closer and closer to 1. Thus, as n increases, it becomes more and more likely that a 6 will be thrown. The question now is, what is the least possible probability of getting at least one six? To answer that question, set n to its lowest possible value, which is 3. In that case, the probability of never getting a 6 is given by:
The probability of getting at least one 6 in three throws is given by:
This value is less than 1/2. As explained earlier, however, as n grows, it becomes ever more likely that at least one throw will yield a 6, so the probability will eventually surpass 1/2.
Thus, Quantity A can be less than or greater than 1/2.
The correct answer is (D): The relationship cannot be determined.
You may have figured intuitively that given three throws, the chance of getting at least one 6 would be \(\frac{3}{6}=\frac{1}{2}\). But if you extend this reasoning, it soon breaks down. If you had six throws, you couldn’t say that you had a 100% chance of rolling a 6, and if you had more than six throws, you certainly wouldn’t want to say the chance was now more than 1. Since a die can keep turning up the same number each time, the events are completely independent, and you can’t just add up the chances.
Answer: D
GMAT-Club-Forum-zhi33urg.png [ 13.74 KiB | Viewed 129 times ]
GMAT-Club-Forum-kobf87l8.png [ 46.48 KiB | Viewed 130 times ]
GMAT-Club-Forum-2blghgyk.png [ 24.6 KiB | Viewed 131 times ]
The easiest way to compute the probability in question is through the “1 − x” shortcut. To do so, imagine the opposite of the event of interest, namely, that none of the n throws yields a 6. The probability of a single throw not yielding a 6 is \(\frac{5}{6}\), and because each throw is independent, the cumulative probability of none of the n throws yielding a 6 is found by multiplication:
Powers of fractions less than one decrease as the exponent increases, so this probability will become very small for large values of n, such that the probability of getting at least one 6 (which is \(1-[\frac{5}{6}]\)) will come closer and closer to 1. Thus, as n increases, it becomes more and more likely that a 6 will be thrown. The question now is, what is the least possible probability of getting at least one six? To answer that question, set n to its lowest possible value, which is 3. In that case, the probability of never getting a 6 is given by:
The probability of getting at least one 6 in three throws is given by:
This value is less than 1/2. As explained earlier, however, as n grows, it becomes ever more likely that at least one throw will yield a 6, so the probability will eventually surpass 1/2.
Thus, Quantity A can be less than or greater than 1/2.
The correct answer is (D): The relationship cannot be determined.
You may have figured intuitively that given three throws, the chance of getting at least one 6 would be \(\frac{3}{6}=\frac{1}{2}\). But if you extend this reasoning, it soon breaks down. If you had six throws, you couldn’t say that you had a 100% chance of rolling a 6, and if you had more than six throws, you certainly wouldn’t want to say the chance was now more than 1. Since a die can keep turning up the same number each time, the events are completely independent, and you can’t just add up the chances.
Answer: D
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GMAT-Club-Forum-zhi33urg.png [ 13.74 KiB | Viewed 129 times ]
Attachment:
GMAT-Club-Forum-kobf87l8.png [ 46.48 KiB | Viewed 130 times ]
Attachment:
GMAT-Club-Forum-2blghgyk.png [ 24.6 KiB | Viewed 131 times ]
