Contains the Center? 5

In general for an $n$ -gon, in order for the center of the circle not to lie inside, we need to have all $n$ points lie within an arc of $180^\circ$ . Let $P_a$ be first point (in clockwise direction) on this arc. Then the probability that the other $n-1$ points lie within $180^\circ$ in clockwise direction from $P_a$ is $(\tfrac12)^{n-1}$ . For point $P_a$ there are $n$ choices. Thus there is probability $n\times (\tfrac12)^{n-1}$ that the center of the circle does not lie inside the $n$ -gon.

Finally, taking the complement, the probability of finding the center inside the $n$ -gon is $1 - n\times (\tfrac12)^{n-1} = \frac{2^{n-1} - n}{2^{n-1}}.$ In the case $n= 7$ , we find $57/64$ .