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Bunuel
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A pair of standard 6-sided dice is rolled once. The sum of the numbers 31 Mar 2019, 21:50
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A pair of standard 6-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?

A. 1/36
B. 1/12
C. 1/6
D. 1/4
E. 5/18
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B
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A pair of standard 6-sided dice is rolled once. The sum of the numbers 01 Apr 2019, 10:41
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Bunuel
A pair of standard 6-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?

A. 1/36
B. 1/12
C. 1/6
D. 1/4
E. 5/18
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pi * r^2<2*pi* r
r<2
so this will be case (1,1),(2,1) & ( 1,2)
total fairs 36
3/36; 1/12
IMO B
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A pair of standard 6-sided dice is rolled once. The sum of the numbers 23 Apr 2019, 01:10
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Archit3110
r<2 is understood.
After that, please explain elaborately.

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A pair of standard 6-sided dice is rolled once. The sum of the numbers 28 Oct 2019, 09:08
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Bunuel
A pair of standard 6-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?

A. 1/36
B. 1/12
C. 1/6
D. 1/4
E. 5/18
Show more

numerical value area circle < numerical value circumference circle
\(πr^2 < 2πr… r^2 < 2r…r^2-2r<0…r(r-2)<0…(r>0)…r-2<0…r<2\)

favorable outcomes: 3 [1,1;1,2;2,1]
\(dices(1,1): d=2, r=1, r^2=1…d>r^2…2>1…answer=yes\)
\(dices(1,2;2,1): d=3, r=1.5, r^2=(3/2)^2=9/4=2.25…d>r^2…3>2.25…answer=yes\)
\(dices(3,3): d=6, r=3, r^2=3^2=9…d>r^2…6>9…answer=no\)
\(dices(3,4; etc…): d=7, r=3.5, r^2=(7/2)^2=49/4=12.25…d>r^2…7>12.25…answer=no\)

total outcomes: \(6^2=36\)

probability: 3/36=1/12

Answer (B)
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A pair of standard 6-sided dice is rolled once. The sum of the numbers 29 Oct 2019, 19:58
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Bunuel
A pair of standard 6-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?

A. 1/36
B. 1/12
C. 1/6
D. 1/4
E. 5/18
Show more

We know that the area of a circle with radius r is πr^2 and the circumference of the same circle is 2πr. In order for the numerical value of the area of the circle to be less than the numerical value of the circle's circumference, we need to have:

πr^2 < 2πr

r < 2

In other words, the diameter has to be less than 4. There are 6 x 6 = 36 outcomes when two dice are rolled, and the outcomes with a sum less than 4 are {1, 1}, {1, 2}, and {2, 1}. So the probability in question is 3/36 = 1/12.

Answer: B
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A pair of standard 6-sided dice is rolled once. The sum of the numbers 29 Nov 2020, 13:17
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Bunuel
A pair of standard 6-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?

A. 1/36
B. 1/12
C. 1/6
D. 1/4
E. 5/18
Show more

We want the d's such that Area < Circumference.
That is \(\pi *r^2 < 2* \pi * r\)
\(r^2 < 2r\)
\(r < 2\). Diameter is twice radius so,
\(d < 4\)
Therefore we want the sum of the rolls to be less than 4, we can only get 2 or 3. There are 36 rolls total and we need to get (1, 1), (2, 1), or (1, 2).
Then the probability is 3 out of 36, which is 1/12.

Ans: B
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A pair of standard 6-sided dice is rolled once. The sum of the numbers 31 Mar 2026, 04:20
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