Before I worked on this problem, it occurred to me that the answers might be different for different methods of randomness.  All the solvers so far, assumed ducks land along a random ray.  Anywhere along the ray is considered the same.

But what if ducks land at random x and y coordinates?  To simulate this, I let x and y be uniform from -1 to +1 and then ignored values that would be outside the circle.  I thought the results would be different, but in fact they were the same:

3 ducks: 0.75

4 ducks: 0.50

For my algorithm to determine if all angles result in all ducks being on the same side:

sort all angles in increasing order;

then subtract the smallest value from all so that one of them is 0;

now there is no confusing subtracting across the 0/360 line;

check each inter-duck angular "distance";

since they are sorted, if any distance is over 180, then they are all on the same side of some diameter.

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import random

import math

def angl(dy,dx):

    if dx == 0:

        if dy ==0:

            return 0

        elif dy > 0:

            return 90

        elif dy < 0:

            return 270

        

    a = math.atan(dy/dx) * 180 / math.pi

    if dy >= 0 and dx >= 0:  # quadrant I

        result =  a

    elif dy >= 0 and dx < 0:  # quadrant II

        result =  a + 180

    elif dy < 0 and dx < 0:  # quadrant III

        result = a + 180

    elif dy < 0 and dx >= 0:  # quadrant IV

        result = a + 360

    return int(result)

def issameside(aList):

    """ bool:  input a list of angles between 0 and 360 return True if they are all on the same side of a diameter.  Algorithm, sort all angles in increasing order then subtract the smallest value from all so that one of them is 0.  then check each inter-duck angular distance.  Since they are sorted, if any distance is over 180 then they are all on the same side of some diameter.    """

    ducks = len(aList)

    locs = sorted(aList)

    locs = [ locs[:][i] - locs[:][0] for i in range(ducks) ]

    distances = []

    for i in range(ducks):

        if i == ducks -1:

            distances.append(360-locs[i])

        else:

            distances.append(locs[i+1]-locs[i])

    if max(distances) >= 180:

        return True

    else:

        return False

reps = 100000

ducks = 3   # or change this to 4

countTotal = 0

countSuccess = 0

for j in range(reps):

    xy_coords = []

    angles = []

    landed = 0

    while landed < (ducks):

        x = (random.uniform(-1,1))

        y = (random.uniform(-1,1))

        if x**2 + y**2 > 1:

            continue

        landed += 1

        xy_coords.append([x,y])

        angles.append(angl(y,x))

    countTotal += 1

    if issameside(angles):

        countSuccess +=1

print(ducks, countSuccess, countTotal, countSuccess/countTotal)