The most probable value (mode) of X ~ B(n, p) satisfies:

• If (n+1)p is NOT an integer: mode = floor((n+1)p)
• If (n+1)p IS an integer: two modes exist — (n+1)p and (n+1)p - 1

Quick check: The mode lies in the interval [(n+1)p - 1, (n+1)p].

Example: n = 6, p = 1/3. Then (n+1)p = 7/3 = 2.33. Since not integer, mode = floor(2.33) = 2. So X = 2 is the most probable number of successes.

Example: n = 5, p = 1/2. Then (n+1)p = 3. Since integer, modes are 3 and 2. Both P(X=2) and P(X=3) are equal and maximum.