Brute force the average number of excess toys to expect from a whole McDonald's set if only half are offered at a time vs. all offered at once.

mc_toy_roller.py

import secrets
from itertools import chain

NUM_TOYS = 10
NUM_TRIALS = 10000


def pick_toy(toys_set):
    return secrets.choice(list(toys_set))


def collect_toys(the_set):
    while any(count < 1 for count in the_set.values()):
        toy = pick_toy(the_set)
        the_set[toy] += 1

def run_single_test():
    # Roll one set until you have at least one of each
    whole_set = {el: 0 for el in range(NUM_TOYS)}
    collect_toys(whole_set)
    # print("WHOLE:" + str(whole_set))

    # Roll first half of set
    second_set_a = {el: 0 for el in range(NUM_TOYS // 2)}
    collect_toys(second_set_a)

    # Roll second half of set
    second_set_b = {el: 0 for el in range(
        NUM_TOYS - len(second_set_a), NUM_TOYS)}
    collect_toys(second_set_b)

    final_set = dict(chain.from_iterable(d.items()
                                         for d in (second_set_a, second_set_b)))
    # print("HALF :" + str(final_set))
    return whole_set, final_set


if __name__ == "__main__":
    whole_excess = 0
    partial_excess = 0
    for _ in range(NUM_TRIALS):
        whole, partial = run_single_test()

        for it in whole:
            whole_excess += whole[it] - 1
        
        for it in partial:
            partial_excess += partial[it] - 1
        
    print("Avg. WHOLE excess: " + str(whole_excess / NUM_TRIALS))
    print("Avg. PART  excess: " + str(partial_excess / NUM_TRIALS))