Alice shuffles an ordinary deck of cards and turns the cards face up one at a time while Bob watches. At any point in this process before the last card is turned up, Bob can guess that the next card is red. Does Bob have a strategy that gives him a probability of success greater that .5?

Let $x_1, x_2, \dots, x_n$ be $n$ points (in that order) on the circumference of a circle. Dana starts at the point $x_1$ and walks to one of the two neighboring points with probability $1/2$ for each. Dana continues to walk in this way, always moving from the present point to one of the two neighboring points with probability $1/2$ for each. Find the probability $p_i$ that the point $x_i$ is the last of the $n$ points to be visited for the first time. In other words, find the probability that when $x_i$ is visited for the first time, all the other points will have already been visited. For instance, $p_1=0$ (when $n>1$), since $x_1$ is the first of the $n$ points to be visited.

Let $\pi$ be a random permutation of $1,2,\dots,n$ (from the uniform distribution). What is the probability that 1 and 2 are in the same cycle of $\pi$?

Choose $n$ points at random (uniformly and independently) on the circumference of a circle. Find the probability $p_n$ that all the points lie on a semicircle. (For instance, $p_1 = p_2 = 1$.) More generally, fix $\theta<2\pi$ and find the probability that the $n$ points lie on an arc subtending an angle $\theta$ .