Mode of Binomial Distribution

The mode is the value of \(X\) that is most likely to occur (with highest probability).

If \(p = 0.5\) and \(n\) is even, there is only one mode at the median position \(\frac{n}{2}\). The exception is when \(p = 0.5\) and \(n\) is odd, then there are two modes which are at \(\frac{n-1}{2}\) and \(\frac{n+1}{2}\).

If \(p \neq 0.5\), one needs to consider the probabilities of values of \(X\) close to \(\mathrm{E}(X)\) and use the GDC table function to check the value of \(X\) that has the highest probability.

When \(p=0.5\), the distribution has only one mode if and only if \(n\) is even.