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29 Nov 2021, 03:54
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Artificial intelligence computer HAL 9000 randomly picks three distinct integers between 1 and 9000, inclusive. If the first number picked is \(a\), the second number picked is \(b\), and the third number picked is \(c\), what is the probability that \(a > b > c\) ?
A. \(\frac{1}{60}\)
B. \(\frac{1}{30}\)
C. \(\frac{1}{20}\)
D. \(\frac{1}{6}\)
E. \(\frac{1}{3}\)
A. \(\frac{1}{60}\)
B. \(\frac{1}{30}\)
C. \(\frac{1}{20}\)
D. \(\frac{1}{6}\)
E. \(\frac{1}{3}\)
29 Nov 2021, 03:54
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Official Solution:
Artificial intelligence computer HAL 9000 randomly picks three distinct integers between 1 and 9000, inclusive. If the first number picked is \(a\), the second number picked is \(b\), and the third number picked is \(c\), what is the probability that \(a > b > c\) ?
A. \(\frac{1}{60}\)
B. \(\frac{1}{30}\)
C. \(\frac{1}{20}\)
D. \(\frac{1}{6}\)
E. \(\frac{1}{3}\)
The question basically asks about the probability that three numbers picked are in decreasing order (\(a > b > c\)).
Any three numbers can be picked in \(3! = 6\) ways, and only one sequence from these 6 will be in decreasing order. So, the probability is \(\frac{1}{6}\).
For example, 2, 5, and 6 can be drawn in 6 ways ({firs number, second number, third number}): {2, 5, 6}, {2, 6, 5}, {5, 2, 6}, {5, 6, 2}, {6, 2, 5} and {6, 5, 2}. Only one from these 6 is in decreasing order: {6, 5, 2}.
Answer: D
Artificial intelligence computer HAL 9000 randomly picks three distinct integers between 1 and 9000, inclusive. If the first number picked is \(a\), the second number picked is \(b\), and the third number picked is \(c\), what is the probability that \(a > b > c\) ?
A. \(\frac{1}{60}\)
B. \(\frac{1}{30}\)
C. \(\frac{1}{20}\)
D. \(\frac{1}{6}\)
E. \(\frac{1}{3}\)
The question basically asks about the probability that three numbers picked are in decreasing order (\(a > b > c\)).
Any three numbers can be picked in \(3! = 6\) ways, and only one sequence from these 6 will be in decreasing order. So, the probability is \(\frac{1}{6}\).
For example, 2, 5, and 6 can be drawn in 6 ways ({firs number, second number, third number}): {2, 5, 6}, {2, 6, 5}, {5, 2, 6}, {5, 6, 2}, {6, 2, 5} and {6, 5, 2}. Only one from these 6 is in decreasing order: {6, 5, 2}.
Answer: D
07 Jun 2024, 17:20
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Interesting question.
I did it this way:
Imagine a line from 1-10 (for simplicity, 1-1000 or whatever would work too).
Pick a random number "a" between 1-10.
Probability you pick something to the left or right of it is 1/2. Lets call that number "b".
So probability that 1---b----a----10 is now 1/2.
Probability that you pick something on the dashes is 1/3. which is also the probability of picking something between 1-b, b-a, a-10. Since you want a>b>c, probability is 1/2 * 1/3 = 1/6.
I did it this way:
Imagine a line from 1-10 (for simplicity, 1-1000 or whatever would work too).
Pick a random number "a" between 1-10.
Probability you pick something to the left or right of it is 1/2. Lets call that number "b".
So probability that 1---b----a----10 is now 1/2.
Probability that you pick something on the dashes is 1/3. which is also the probability of picking something between 1-b, b-a, a-10. Since you want a>b>c, probability is 1/2 * 1/3 = 1/6.
