The following question is obtained at Android's app probability puzzle.
Imagine a country in which 10 people have voted in an election opposing a woman named C and a demogogue named T. Suppose C received 7 votes, while 3 voted for candidate T. - but the public doesn't know that yet because the votes have not been counted.
The voters' ballots are placed in a large urm, Before it is opened, the urn is shaken in a way that randomly shuffles the ballots, with all orderings equally likely. The urn is then opened and the ballots taken out one by one and tallied. Given that 7 people voted for C (and 3 for T), we know she will end up with a net advantage of 4 votes as the end of the count. There is randomness in how we get there, however.
What is the probability that C is strictly ahead of T at every point in the counting process?
Let C vote be $+1$ and T vote be $-1.$ So, the question is asking the probability that a random walk which starts at origin that does not cross below $y = 1.$
I tried to use Catalan number to solve the question. However, I am not able to do so as Catalan number gives the number of paths that do not cross below x-axis.
The answer is $0.4.$
After the hint given by @Hetebrij, I think I can solve it.
The first vote must be C with probability $\frac{7}{10}.$ So, we are at $y=1$ at time $1$. There are 6 votes of C and 3 votes of T remaining. By applying similar reasoning as in Catalan number, we have $$\frac{\binom{9}{3} - \binom{9}{2}}{\binom{9}{3}} = \frac{48}{84}$$ where the numerator is calculated based on the total number of paths minus the total number of paths that will cross below $y=1$, where I use the reflection principle to calculate the latter. Multiplying the fraction with $\frac{7}{10}$ leads to $0.4.$