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  4. A pair of standard 6-sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. The probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference, is

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Q. A pair of standard 6-sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. The probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference, is

Probability - Part 2

Solution:

$\frac{\pi d ^2}{4}<\pi d \Rightarrow d ^2<4 d \Rightarrow d ( d -4)<0 \Rightarrow d \in[1,3]$
but sum of two dice $\neq 1$
$\therefore d =2$ or 3
Now, $P(2$ or 3$)=\frac{1}{36}+\frac{2}{36}=\frac{3}{36}=\frac{1}{12}$.