Why is the probability that Kip could go back and kill his grandfather  $10^{-10^{60}}$? In The Universe In A Nutshell, page 153, Hawking says that 

The probability that Kip could go back and kill his grandfather is $10^{-10^{60}}$. 

How do we get this result?
 A: There is no working quantum theory of gravity, however one of the approaches physicists have played with is a sum over paths approach. In quantum electrodynamics you can describe the behaviour of e.g. an electron as a sum over all the possible paths that electron could take (see the link for more info). In an analogous way you could describe the evolution of spacetime curvature as a sum over all possible spacetime curvatures.
Most of these paths through spacetime curvature will be well behaved, much like the well behaved spacetime we see around us. However some of the paths will contain esoteric structures like wormholes that allow closed time-like curves i.e. time travel. When you do a sum over paths you weight each path by a probability, and we expect the esoteric paths to have a very low probability so it's very unlikely you could encounter such a path and travel in time. However the probability is not zero.
I'm not familiar with this bit of Hawking's work, but from the book it appears he's taking a model spacetime that is relatively (no pun intended!) easy to describe using a sum over paths, and using that as a model for a realistic spacetime. His calculation then gives the probability of a path that would take Kip Thorne back in time to meet his grandfather as 1 in $10^{10^{60}}$.
I would take this result with a pinch of salt (as I'm sure Hawking would agree)!
For a more detailed description see this article from the Dept of Applied Mathematics and Theoretical Physics at Cambridge. Hawking was (is?) a professor in this department.
A: Note that whatever is the specific mechanism that Hawking proposes, $10^{-60}$ is not very far away of the probability that a spontaneous identical copy of a person will appear in a specific spot of space and time, just out of random fluctuations. So one could say that a lot larger figure can be derived rather directly from non-relativistic statistical physics
