Download FI_The Monty Hall

Author: Raudaskyla Christian College
Monty Hall - Problem
Introduction
In some TV-shows the winner may choose one of three doors. Behind one door there is the top prize, for
example a trip or a car. There is nothing behind two other doors. At first player chooses one door. After that
the host of this game show opens one of the two doors player has not chosen. Behind this door, there is
nothing. Then the host asks: ”Do you want to change the door?” This is the main point of Monty Hall –
problem: does it pay off to change? The host knows all the time, where the prize is.
Theory
This interesting problem is based on probability. Results in theory can easily be shown through a table. Here
we assume that player picks the first door at start:
Chosen door
Door 2
Door 3
Result by changing
Prize
Empty
Empty
Empty
Prize
Empty
Empty
Empty
Prize
Lose
Win
Win
Result, if player
doesn’t change
Win
Lose
Lose
Thus this table shows that if you change the door, the probability to win is 2/3. If you don’t change the door,
the probability to win is 1/3.
In other words: by changing the door player loses only if he has chosen right at first. Probability of that is
1/3. Therefore probability to win by changing is 2/3.
Organization
We tested this problem in mathematics lesson at Raudaskylän Lukio. The group consisted of first and second
year students. They were divided in small groups (3-4 students/group). Each group had three cups and one
”prize”, for example a rubber. In a group, student took roles as a host and as a player. The roles were
changed. They listed their statistics by changing and by staying in first option.
Results
When each group had finished their job, we got following results (presented at the next page):
Group 1
Group 2
Group 3
Group 4
Group 5
Group 6
Total
Wins
14
16
13
16
15
18
Change
Defeats
11
9
12
9
10
7
Wins
12
8
9
7
8
11
No change
Defeats
13
17
16
18
17
14
92
58
55
95
When player changed the door, his/her winning percentage was 61,3. Without a change the percentage was
36,7.
Conclusion
Our experimental game supports the theory. Neverheless, this problem is also psychological and after first
phase player may think that chances are now even: two doors and there is a prize behind one or the other.
Dare I change? This test and probability theory shows that the change pays off. As we all know ”variety is
the spice of life!”