The answer turns out to be 10%. My intuition was that the probability would be very small.
In the general case where A gets p votes and B gets q votes, the chances of keeping the lead is the same as the winning percentage, (p – q)/(p+q). Here is the Wikipedia article on it.
There is no intuitive way that I can think of as to why the formula is true, although if you know basic combinatorics, the mathematics is not too bad using the reflection method. This article gives several proofs, only the first of which I bothered with. Unlike the Wikipedia article, it includes a useful picture showing how the reflection principle is used. You might find the reasoning of interest even if you can’t follow all the math.
Here is a brief summary.
As @zenvelo points out, A has to get the first vote. We first compute the total number of outcomes where A gets that first vote.
Then an interesting graph is produced, showing A’s lead after each vote, which will be 0 if the vote is tied and negative if B takes the lead.
We want to subtract all the cases where A gets the first vote, but the vote graph touches the 0 line. Using the graph, the article shows that this is exactly the same as the total number of cases where the final vote is the same but where B gets the first vote. We can compute this number and subtract from the first and, after a little algebraic manipulation (not shown), end up with the final answer.