Grace rolls two fair six-sided dice and adds the results. She than draws a square that has her result as the length of the diagonal. What is the probability that the numerical value of the area of her square will be less than the numerical value of the perimeter? The answer is 5/18.
I drew a table and added up the results of dice rolls. But I am not quite sure of the length of the diagonal? Please help. Thank you in advance.
If the length of the diagonal is $d$, the side length will be $\frac{d}{\sqrt{2}}$. So the area will be $\frac{d^{2}}{2}$ and the perimeter will be $\frac{4d}{\sqrt{2}} = 2d\sqrt{2}$.
For the area to be less than the perimeter, the condition is
$$ \frac{d^{2}}{2} < 2d\sqrt{2} $$
Which can be rearranged
$$ d^{2} < 4d\sqrt{2} $$
$$ d^{2} - 4d\sqrt{2} < 0 $$
$$ d(d - 4\sqrt{2} ) < 0 $$
So $0 < d < 4\sqrt{2}$
So $d \le 5$ for the integer sum on the pair of dice.
This gives a probability of $\frac{5}{18}$ from your table.
If the result is $x$, then the area is $\displaystyle \frac{1}{2}x^2$ and the perimeter is $2\sqrt{2}x$.
$\displaystyle \frac{1}{2}x^2<2\sqrt{2}x$ $\Longleftrightarrow $ $\displaystyle x<4\sqrt{2}$ $\Longleftrightarrow $ $x\le 5$
The possible outcomes of the dice are $(1,1),(1.2),(1,3),(1,4),(2,1),(2,2),(2,3),(3,1),(3,2),(4,1)$.
The probability is $\displaystyle \frac{10}{36}=\frac{5}{18}$.
