Survival of the Rabbit!

Suppose that the rabbit is at a distance $r$ metres from the centre $O$ of the table. The rabbit will not fall off provided that it heads of in a direction within the minor segment $RAB$ . The (conditional) probability of this happening is $\tfrac{1}{\pi}\cos^{-1}\big(\tfrac12r\big)$ .

Thus the probability that the rabbit does not fall off the table is $\begin{array}{rcl} \displaystyle \tfrac{1}{\pi}\int_0^1 \tfrac{1}{\pi}\cos^{-1}\big(\tfrac12r\big) 2\pi r\,dr & = & \displaystyle \tfrac{2}{\pi}\int_0^1 r\cos^{-1}\big(\tfrac12r\big)\,dr \\ & =& \displaystyle \tfrac{2}{\pi}\Big[\tfrac12r^2 \cos^{-1}\big(\tfrac12r\big)\Big]_0^1 + \tfrac{1}{\pi}\int_0^1 \frac{r^2}{\sqrt{4-r^2}}\,dr \\ & = & \displaystyle\tfrac13 + \tfrac{1}{\pi}\int_0^{\frac16\pi} \frac{4\sin^2\theta}{2\cos\theta} 2\cos\theta\,d\theta \\ & = & \displaystyle \tfrac13 + \tfrac{2}{\pi}\int_0^{\frac16\pi}(1 - \cos2\theta)\,d\theta \\ & = & \displaystyle \tfrac13 + \tfrac{2}{\pi}\Big[\theta - \tfrac12\sin2\theta\Big]_0^{\frac16\pi} \\ & = & \tfrac23 - \tfrac{\sqrt{3}}{2\pi} \; = \; \boxed{0.391} \end{array}$

lets define a function that has the following properties;

p(r) equals to the area contained inside if the sub-circle with radius r divided by the entire area of the table.

here p(r) is the probability that the rabbit is put inside of a circle with radius r<=1, however we want to calculate the chance if the rabbit is put on a circumference of thickness $dr \rightarrow 0$ .

$P(r)=p(r) -p(r-dr)$

$P(r)=\frac{\pi r^2}{\pi R^2} -\frac{\pi (r-dr)^2}{\pi R^2}$

$P(r)= \frac{\pi 2rdr}{\pi R^2}= 2rdr$

Now the interesting part ( Geometry part of this problem )

using the cosine law we can simply yield that $\phi =acos(\frac{r}{2})$

the problem here that if the rabbit was put inside he can choose a wrong direction and fall off the table so

the chance that this idiot choose the right direction is $C(r)=\phi(r)/\pi$

and the total chance that he wont fall off is the sum of $C(r)*P(r)$

So what we want to calculate is the chance he still be on the table W so

$W=∫_0 ^1 (\frac{2}{\pi}r acos(\frac{r}{2})dr ) =**0.391**$

• if you want to know how the last integral yielded click here