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Try this beautiful problem from Probability based on dice Problem

## Dice Problem- AMC-10A, 2011- Problem 14

A pair of standard 6-sided fair 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?

• $\frac{1}{12}$
• $\frac{7}{12}$
• $\frac{5}{12}$
• $\frac{1}{2}$
• $\frac{1}{9}$

### Key Concepts

Probability

dice

circle

But try the problem first…

Answer: $\frac{1}{12}$

Source

AMC-10A (2011) Problem 14

Pre College Mathematics

## Try with Hints

First hint

Given that A pair of standard 6-sided fair dice are rolled once. The sum of the numbers rolled determines the diameter of a circle. The numerical value of the area of the circle is less than the numerical value of the circle’s circumference. Let the radius of the circle is $r$. Then the area of the circle be $\pi(r)^2$ and  circumference be $2\pi r$

can you finish the problem……..

Second Hint

Now according to the given condition we say that $\pi(r)^2 <2\pi{r}$$\Rightarrow r<2$

As The sum of the numbers rolled determines the diameter of a circle, therefore $r<2$ then the dice must show $(1,1)$,$(1,2)$,$(2,1)$

can you finish the problem……..

Final Step

Therefore there $3$ choices out of a total possible of $6 \times 6 =36$, so the probability is $\frac{3}{36}=\frac{1}{12}$