One hundred students are divided into two equal groups. Both groups vote (yes or no). Find the probability that the groups 1 and 2 both reach a majority.
Students vote independently and the probability for each student to vote yes ½. A majority can be achieved with either yes or no votes.
The probability of a tie is $$P_{tie}=\frac{\binom{50}{25}}{2^{50}}$$ thus the probability that neither group reaches a tie is $$P = (1-P_{tie})^2$$ (where all events are assumed to be independent). This gives $$P = \left(1 - \frac{\binom{50}{25}}{2^{50}}\right)^2 \approx \left(1 - \frac{1}{5\sqrt{\pi}}\right)^2\approx 0.79$$ where in the second step we approximated using Stirling's formula.