Given that the distribution is bimodal,
If one mode is X = k, then the other mode will be X = k - 1 or k + 1. We would expect the modes to be next to each other since the distribution is bell shaped. i.e highest probabilities are next to each other.


Notice that (k,k +1) and (k -1,k) are interchangeable since what's most important is the are apart by 1. As the question already gave us a formula we will use (k,k -1) .

As the modes have equal probabilities,

p(k) = p(k - 1)

So (n - k + 1)p = k(1 - p)

(n + 1)p - kp = k - kp

(n + 1)p = k

p = k/(n + 1)

18/25 = k/(n+1)

k and n are integers. So n + 1 is also an integer.

Since n > 0 (you can't have 0 trials) ,
0 > k ≥ n, ( For this case k ≠ 0 since the LHS ≠ 0, so numerator cannot be 0. And maximum value of k can only be equal to n since there are only n trials, eg. can't be having success for 21 trials if there's only 20 trials done.)

Then n + 1 > k , n + 1 > 1

Therefore k/(n+1) is a proper fraction since denominator is greater than 1 and numerator is smaller than the denominator.


As the fraction 18/25 is in its simplest form, we can say that :

Least value of n + 1 is the numerator value = 25, and correspondingly the value of k is 18, the denominator value.

(Fractions like 36/50, 54/75, 72/100 etc are possible too but we want the least value so we look for the simplest form)

Least value of n = 24
And k - 1 = 17

so the corresponding modes are when X = 17 and X = 18

you can visit

https://homepage.divms.uiowa.edu/~mbognar/applets/bin.html and try to simulate the distribution for visualising