The Howie Mandel Problem

Now imagine you are on Deal or No Deal, a television show in which there are 26 cases filled with different amounts of cash ranging from a penny to a million dollars.  You start out by picking one case.  Then for each round, the host, Howie Mandel, asks you to eliminate one of the leftover cases, which is opened to reveal the amount inside, after which a mysterious banker offers to buy your case for an amount that varies based on the amounts that are leftover.  You then decide on either taking the money (deal) or playing another round in hopes of getting a better offer from the banker (no deal).

On the rare occasion that you actually get it down to just two cases left in which one of those two cases has the million dollars, Howie will offer you the chance to switch cases.  What should you do?

This problem is similar to the “Monty Hall Problem”, where you are a contestant on Let’s Make a Deal and Monty Hall shows you three doors.  Behind one door is a brand new car, but behind the other two doors are two goats.  You pick a door, but just before it is opened, Monty says he will open a different door other than the door that you picked that has a goat behind it, and then offers you the chance to switch doors.  As proved in a previous blog, you should switch doors because switching gives you a 2 in 3 chance of winning (since your original choice was a 1 in 3 chance, it leaves you with a 2 in 3 chance for the remaining doors, in which one choice is eliminated).

So is the “Howie Mandel Problem” similar enough to the “Monty Hall Problem” that you should switch cases?  Or are the problems different enough so that it doesn’t matter if you switch cases or not, because either case has a 1 in 2 chance of winning?  Once again, I wrote a small computer program in Python that simulated the scenario one million times and kept track of how many times you would win by switching cases out of how many times you would actually be in a situation where you only have two cases left.  The computer program is as follows:

01	#The Howie Mandel Solution
02	#Python 2.7.3
03
04	#Be able to pick a random number later.
05	import random
06
07	#Total number of trials to calculate.  The higher the number, the
08	#  more accurate the probability solution.
09	trials = 1000000
10
11	#Total number of cases
12	cases = 26
13
14	#Keep track of the number of trials which get down to having two
15	#  cases at the end, originally starting at zero.
16	trials_down_to_two_cases = 0
17
18	#Keep track of the number of wins if you switch cases at the end,
19	#  originally starting at zero.
20	trials_won_by_switching = 0
21
22	#Go through each trial one by one.
23	for trial in range(1, trials + 1):
24
25	#Randomly pick a number between 1 and the number of cases
26	#  for the case with the million dollars.
27	case_with_big_money = random.choice(range(1, cases))
28
29	#Randomly pick a number between 1 and the number of cases
30	#  for the case you choose first.
31	case_you_choose_first = random.choice(range(1, cases))
32
33	#Randomly pick a number between 1 and the number of cases
34	#  (excluding the case you already chose) for the case
35	#  you leave for last.
36	case_you_leave_for_last = random.choice(range(
37	1, case_you_choose_first - 1) + range(
38	case_you_choose_first + 1, cases))
39
40	#If the case with the million dollars is in either the case
41	#  you choose first or the case you save for last,
42	#  then the game will get down to only two cases left, so add
43	#  one to the number of trials down to two cases.
44	if (case_with_big_money == 1 or case_with_big_money == 2):
45	trials_down_to_two_cases += 1
46
47	#If case with the million dollars is in the case you save for
48	#  last, then add one to the number of trials won by switching cases.
49	if (case_you_leave_for_last == case_with_big_money):
50	trials_won_by_switching += 1
51
52	#When all trials have been run, print the number of trials won
53	#  by switching and give the percent.
54	print "Total number of times it will get down to two cases: " + str(
55	trials_down_to_two_cases) + " out of " + str(trials) + " (" + str(
56	int(round(trials_down_to_two_cases * 100.0 / trials))) + "%)"
57	print "Total number of wins by switching: " + str(
58	trials_won_by_switching) + " out of " + str(
59	trials_down_to_two_cases) + " (" + str(int(round(
60	trials_won_by_switching * 100.0 / trials_down_to_two_cases))) + "%)"

The outcome after running the computer program is as follows:

>>	Total number of times it will get down to two cases: 80083 out of 1000000 (8%)
>>	Total number of wins by switching: 39989 out of 80083 (50%)

Therefore, it makes no difference whether you switch cases or not, because either case has a 1 in 2 chance of winning.  The difference between the “Howie Mandel Problem” and the “Monty Hall Problem” is that in the “Howie Mandel Problem”, the options are being randomly eliminated by the contestant, but in the “Monty Hall Problem”, the options are knowingly eliminated by the host.  That tiny detail makes all the difference!