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The first time I heard about the Monty Hall problem was while watching TV. The host of the TV show had presented the problem as you typically see. I am recapping it for your benefit now.
The TV show has three doors. Behind one of which there is a prize. The guest is expected to choose one of the doors. Upon the first selection the host opens up another door to reveal that there is no prize behind it. The dilemma facing the guest now is; should he change his original selection? Or stick with it. Everybody has heard some variation of this problem and it did cause quite some controversy when it first came out. It has since been proven that you should always switch the door. This presents an interesting problem, because what changed? Why would you suddenly want to switch the door?
The fact is you have new information, which should dictate your choice. Simply speaking, your probability will change between the first and the second run because of the fact that you have additional information. Now rather than try to prove this mathematically, I’m just going to go ahead and explain it another way. Say for example, you always put the prize behind door 1. After a couple of attempts, you start to figure out the pattern and you understand that the prize will always be behind door 1. This additional information. Should be taken into account when making your guess. While the probability remains 1/3 because you have three doors and only one of them can have a prize behind it. The additional information that the prize is always behind door one can change your probability from 1/3 to 100%.
This aspect of additional information changing the probability was something that wasn’t taken into account when this problem was first introduced. A number of famous mathematicians had ridiculed the original author (Vos Savant) who suggested that you should always switch. Subsequently, people have investigated this further and proven that you should always switch. Computer simulations have also done the same.
In this post I present one more computer simulation that you can do using Microsoft SQL Server.
SET NOCOUNT ON -- improves loop performace
-- drop temp table used to store results
IF EXISTS (
SELECT 1
FROM tempdb.INFORMATION_SCHEMA.TABLES
WHERE TABLE_NAME LIKE '%Mont%'
)
BEGIN
DROP TABLE #MontyHall
END
-- create table to store results for montyhall choices
CREATE TABLE #MontyHall (id INT identity(1, 1), runid VARCHAR(100), cup0 BIT, cup1 BIT, cup2 BIT, usercup INT, firstrunwinner INT, RUndescription VARCHAR(100))
DECLARE @cntr INT = 0
-- loop a 10000 runs of the game
WHILE @cntr <= 10000
BEGIN
--- declare our three cups
DECLARE @cup0 BIT = 0
DECLARE @cup1 BIT = 0
DECLARE @cup2 BIT = 0
-- run id to correlate first and second choices for the case when a switch happens
DECLARE @runid VARCHAR(100) = cast(newid() AS VARCHAR(100))
-- assigns a random number between 0 and 2 to simulate a guess of one of the cups -- this is the winning cup
DECLARE @cupselect INT = (
SELECT cast(round(rand() * 10, 0) AS INT) % 3
)
-- a step to set the value of cup vairable based on what was the value of the @cupselect variable
IF @cupselect = 0
SET @cup0 = 1
IF @cupselect = 1
SET @cup1 = 1
IF @cupselect = 2
SET @cup2 = 1
--- simulates a guess by the user ( select 0,1 or 2 as the cup choices )
DECLARE @cupuser INT = (
SELECT cast(round(rand() * 10, 0) AS INT) % 3
)
-- In the first run the winner is already decicded after one run becuase the user never changes the inital selection
-- storing the results to confirm the same and verify if the probability is 1/3
INSERT INTO #Montyhall ([runid], [cup0], [cup1], [cup2], [usercup], [firstrunwinner], RUndescription)
SELECT @runid, @cup0 AS cup0, @cup1 AS cup1, @cup2 AS cup2, @cupuser AS UserCup, @cupselect AS WinnerFirstRound, 'First Chance'
--identify empty cup
-- empty cup can only be the cup that doesnt contain the winner or the users original choice.
-- the empty cup is the one that is revealed. Forcing the user to take a chance of sticking with the original
-- choice or switching the cup that wasnt revealed without knowing if they got it right the first time.
-- in the second we simulate the user alwats switches
DECLARE @emptycutp INT = (
SELECT TOP 1 cup
FROM (
SELECT cup
FROM (
SELECT 0 AS cup
UNION
SELECT 1
UNION
SELECT 2
) a -- order by newid()
EXCEPT -- ignore the below two and choose what is left.
SELECT *
FROM (
SELECT @cupuser AS cup -- users choice
UNION
SELECT @cupselect -- winning up
) b
) q
ORDER BY newid()
)
-- Set empty cup as NULL just so that its easier to identify within the table.
IF @emptycutp = 0
SET @cup0 = NULL
IF @emptycutp = 1
SET @cup1 = NULL
IF @emptycutp = 2
SET @cup2 = NULL
-- set the users choice to always select the cup that was not revealed
SELECT @cupuser = (
SELECT cup
FROM (
SELECT 0 AS cup
UNION
SELECT 1
UNION
SELECT 2
) a
EXCEPT
SELECT *
FROM (
SELECT @cupuser AS cup -- new choice cannot be the orignal choice
UNION
SELECT @emptycutp -- or the empty cup that was revealed.
) aa
)
INSERT INTO #Montyhall ([runid], [cup0], [cup1], [cup2], [usercup], [firstrunwinner], RUndescription)
SELECT @runid, @cup0 AS cup0, @cup1 AS cup1, @cup2 AS cup2, @cupuser AS UserCup, @cupselect AS WinnerFirstRound, 'Second Chance'
SET @cntr += 1
END
SELECT RUndescription, Winstate ,count(*) as '#wins'
FROM (
SELECT *, WinState = CASE
WHEN [firstrunwinner] = [usercup]
THEN 'Winner'
ELSE 'Loser'
END
FROM #MontyHall
) a
GROUP BY RUndescription, Winstate
ORDER BY RUndescription, Winstate
select *,WinState = CASE
WHEN [firstrunwinner] = [usercup]
THEN 'Winner'
ELSE 'Loser'
END from #MontyHall
order by runid