1. The Monty Hall Problem
A bit of the Monty Hall Show(a bit controversial):
http://en.wikipedia.org/wiki/Monty_Hall
The Hollywood Version:
http://www.youtube.com/watch?v=cXqDIFUB7YU
Wikipedia Version:
http://en.wikipedia.org/wiki/Monty_Hall_problem
A Game Simulator:
http://math.ucsd.edu/~crypto/Monty/Montytitle.html
© Frank Ma 2012

2. The Monty Hall Problem

3. The Monty Hall Problem
The Monty Hall problem got its name from the TV game show
Let’s Make a Deal hosted by Monty Hall.

4. The Monty Hall Problem
The Monty Hall problem got its name from the TV game show
Let’s Make a Deal hosted by Monty Hall.
A similar game is shown here.

5. The Monty Hall Problem
The Monty Hall problem got its name from the TV game show
Let’s Make a Deal hosted by Monty Hall.
A similar game is shown here.
a. Monty presents to us three sealed and labeled boxes.
1 2 3

6. The Monty Hall Problem
The Monty Hall problem got its name from the TV game show
Let’s Make a Deal hosted by Monty Hall.
A similar game is shown here.
a. Monty presents to us three sealed and labeled boxes.
One contains a $100 bill and the other two contain nothing.
1 2 3

7. The Monty Hall Problem
The Monty Hall problem got its name from the TV game show
Let’s Make a Deal hosted by Monty Hall.
A similar game is shown here.
a. Monty presents to us three sealed and labeled boxes.
One contains a $100 bill and the other two contain nothing.
b. We select a box to keep but we are not allowed to open it
so we don‟t know what‟s inside.
1 2 3

8. The Monty Hall Problem
The Monty Hall problem got its name from the TV game show
Let’s Make a Deal hosted by Monty Hall.
A similar game is shown here.
a. Monty presents to us three sealed and labeled boxes.
One contains a $100 bill and the other two contain nothing.
b. We select a box to keep but we are not allowed to open it
so we don‟t know what‟s inside. Let‟s say we selected box 2.
selected box
1 2 3

9. The Monty Hall Problem
The Monty Hall problem got its name from the TV game show
Let’s Make a Deal hosted by Monty Hall.
A similar game is shown here.
a. Monty presents to us three sealed and labeled boxes.
One contains a $100 bill and the other two contain nothing.
b. We select a box to keep but we are not allowed to open it
so we don‟t know what‟s inside. Let‟s say we selected box 2.
One of these or maybe both are empty.
selected box
1 2 3

10. The Monty Hall Problem
The Monty Hall problem got its name from the TV game show
Let’s Make a Deal hosted by Monty Hall.
A similar game is shown here.
a. Monty presents to us three sealed and labeled boxes.
One contains a $100 bill and the other two contain nothing.
b. We select a box to keep but we are not allowed to open it
so we don‟t know what‟s inside. Let‟s say we selected box 2.
c. Monty shows an empty box to us from one of the remaining.
two boxes.
One of these or maybe both are empty.
selected box
1 2 3

11. The Monty Hall Problem
The Monty Hall problem got its name from the TV game show
Let’s Make a Deal hosted by Monty Hall.
A similar game is shown here.
a. Monty presents to us three sealed and labeled boxes.
One contains a $100 bill and the other two contain nothing.
b. We select a box to keep but we are not allowed to open it
so we don‟t know what‟s inside. Let‟s say we selected box 2.
c. Monty shows an empty box to us from one of the remaining.
two boxes. Let‟s say he shows us that box1 is empty.*
1 2 3

12. The Monty Hall Problem
The Monty Hall problem got its name from the TV game show
Let’s Make a Deal hosted by Monty Hall.
A similar game is shown here.
a. Monty presents to us three sealed and labeled boxes.
One contains a $100 bill and the other two contain nothing.
b. We select a box to keep but we are not allowed to open it
so we don‟t know what‟s inside. Let‟s say we selected box 2.
c. Monty shows an empty box to us from one of the remaining.
two boxes. Let‟s say he shows us that box1 is empty.*
* Which box is opened depends on where the $100 is.
Monty might have to open box3 because the $100 is in box1.
We‟re just saying that in this case that Monty opens box1.
1 2 3

13. The Monty Hall Problem
The Monty Hall problem got its name from the TV game show
Let’s Make a Deal hosted by Monty Hall.
A similar game is shown here.
a. Monty presents to us three sealed and labeled boxes.
One contains a $100 bill and the other two contain nothing.
b. We select a box to keep but we are not allowed to open it
so we don‟t know what‟s inside. Let‟s say we selected box 2.
c. Monty shows an empty box to us from one of the remaining.
two boxes. Let‟s say he shows us that box1 is empty.
d. Now Monty asks us “Do you want to trade your box with the
remaining one (in this case box 3)*?”.
1 2 3

14. The Monty Hall Problem
The Monty Hall problem got its name from the TV game show
Let’s Make a Deal hosted by Monty Hall.
A similar game is shown here.
a. Monty presents to us three sealed and labeled boxes.
One contains a $100 bill and the other two contain nothing.
b. We select a box to keep but we are not allowed to open it
so we don‟t know what‟s inside. Let‟s say we selected box 2.
c. Monty shows an empty box to us from one of the remaining.
two boxes. Let‟s say he shows us that box1 is empty.
d. Now Monty asks us “Do you want to trade your box with the
remaining one (in this case box 3)*?”.
*If he had opened box3 instead, then he would be asking us
that if we want to trade the chosen box2 with box1.
1 2 3

15. The Monty Hall Problem
The Monty Hall problem got its name from the TV game show
Let’s Make a Deal hosted by Monty Hall.
A similar game is shown here.
a. Monty presents to us three sealed and labeled boxes.
One contains a $100 bill and the other two contain nothing.
b. We select a box to keep but we are not allowed to open it
so we don‟t know what‟s inside. Let‟s say we selected box 2.
c. Monty shows an empty box to us from one of the remaining.
two boxes. Let‟s say he shows us that box1 is empty.
d. Now Monty asks us “Do you want to trade your box with the
remaining one (in this case box 3)*?”.
This is the Monty Hall problem,
1 2 3

16. The Monty Hall Problem
The Monty Hall problem got its name from the TV game show
Let’s Make a Deal hosted by Monty Hall.
A similar game is shown here.
a. Monty presents to us three sealed and labeled boxes.
One contains a $100 bill and the other two contain nothing.
b. We select a box to keep but we are not allowed to open it
so we don‟t know what‟s inside. Let‟s say we selected box 2.
c. Monty shows an empty box to us from one of the remaining.
two boxes. Let‟s say he shows us that box1 is empty.
d. Now Monty asks us “Do you want to trade your box with the
remaining one (in this case box 3)*?”.
This is the Monty Hall problem,
that is, should we change in the mid–stream?
1 2 3

17. The Monty Hall Problem
To most people, this is a bizarre proposition
1 2 3

18. The Monty Hall Problem
To most people, this is a bizarre proposition
Its here.
1 2
100 3

19. The Monty Hall Problem
To most people, this is a bizarre proposition
or here.
1 2 3
100

20. The Monty Hall Problem
To most people, this is a bizarre proposition
here
1 2
100 3

21. The Monty Hall Problem
To most people, this is a bizarre proposition
or here.
1 2 3
100

22. The Monty Hall Problem
To most people, this is a bizarre proposition - they reason that
box 2 and box 3 each has ½ chance of containing the $100,
so of course it makes no difference whether we trade or not.
or here.
1 2 3
100

23. The Monty Hall Problem
To most people, this is a bizarre proposition - they reason that
box 2 and box 3 each has ½ chance of containing the $100,
so of course it makes no difference whether we trade or not.
Surprisingly the right action is to trade away our 1st choice.
1 2 3

24. The Monty Hall Problem
To most people, this is a bizarre proposition - they reason that
box 2 and box 3 each has ½ chance of containing the $100,
so of course it makes no difference whether we trade or not.
Surprisingly the right action is to trade away our 1st choice.
Fact: By changing our 1st choice in the middle of the
Monty Hall game we double the chance of obtaining the
$100 from 1/3 to 2/3.
1 2 3

25. The Monty Hall Problem
To most people, this is a bizarre proposition - they reason that
box 2 and box 3 each has ½ chance of containing the $100,
so of course it makes no difference whether we trade or not.
Surprisingly the right action is to trade away our 1st choice.
Fact: By changing our 1st choice in the middle of the
Monty Hall game we double the chance of obtaining the
$100 from 1/3 to 2/3.
In the rest of this presentation, we‟ll
• justify this fact
• clarify the misconception
1 2 3

26. The Monty Hall Problem
Verification of the Monty Hall Fact

27. The Monty Hall Problem
Verification of the Monty Hall Fact
We want to verify that changing our original choice in the
middle of the game actually doubled the chance of winning.

28. The Monty Hall Problem
Verification of the Monty Hall Fact
We want to verify that changing our original choice in the
middle of the game actually doubled the chance of winning.
We‟ll say we win (W) if we obtain the $100 at the end and we
lose (L) if we obtain an empty box at the end.

29. The Monty Hall Problem
Verification of the Monty Hall Fact
We want to verify that changing our original choice in the
middle of the game actually doubled the chance of winning.
We‟ll say we win (W) if we obtain the $100 at the end and we
lose (L) if we obtain an empty box at the end.
Let„s review the game.

30. The Monty Hall Problem
Verification of the Monty Hall Fact
We want to verify that changing our original choice in the
middle of the game actually doubled the chance of winning.
We‟ll say we win (W) if we obtain the $100 at the end and we
lose (L) if we obtain an empty box at the end.
Let„s review the game.
a. Monty presents to us three sealed and labeled boxes.
1 2 3

31. The Monty Hall Problem
Verification of the Monty Hall Fact
We want to verify that changing our original choice in the
middle of the game actually doubled the chance of winning.
We‟ll say we win (W) if we obtain the $100 at the end and we
lose (L) if we obtain an empty box at the end.
Let„s review the game.
a. Monty presents to us three sealed and labeled boxes.
One contains a $100 bill and the other two contain nothing.
1 2 3

32. The Monty Hall Problem
Verification of the Monty Hall Fact
We want to verify that changing our original choice in the
middle of the game actually doubled the chance of winning.
We‟ll say we win (W) if we obtain the $100 at the end and we
lose (L) if we obtain an empty box at the end.
Let„s review the game.
a. Monty presents to us three sealed and labeled boxes.
One contains a $100 bill and the other two contain nothing.
Let‟s say we select box 2 to keep.
1 2 3

33. The Monty Hall Problem
Verification of the Monty Hall Fact
We want to verify that changing our original choice in the
middle of the game actually doubled the chance of winning.
We‟ll say we win (W) if we obtain the $100 at the end and we
lose (L) if we obtain an empty box at the end.
Let„s review the game.
a. Monty presents to us three sealed and labeled boxes.
One contains a $100 bill and the other two contain nothing.
Let‟s say we select box 2 to keep.
1 2 3

34. The Monty Hall Problem
Verification of the Monty Hall Fact
We want to verify that changing our original choice in the
middle of the game actually doubled the chance of winning.
We‟ll say we win (W) if we obtain the $100 at the end and we
lose (L) if we obtain an empty box at the end.
Let„s review the game.
a. Monty presents to us three sealed and labeled boxes.
One contains a $100 bill and the other two contain nothing.
Let‟s say we select box 2 to keep.
c. Monty shows an empty box.
1 2 3

35. The Monty Hall Problem
Verification of the Monty Hall Fact
We want to verify that changing our original choice in the
middle of the game actually doubled the chance of winning.
We‟ll say we win (W) if we obtain the $100 at the end and we
lose (L) if we obtain an empty box at the end.
Let„s review the game.
a. Monty presents to us three sealed and labeled boxes.
One contains a $100 bill and the other two contain nothing.
Let‟s say we select box 2 to keep.
c. Monty shows an empty box. Then Monty gives us the option
to switch boxes.
1 2 3

36. The Monty Hall Problem
Verification of the Monty Hall Fact
We want to verify that changing our original choice in the
middle of the game actually doubled the chance of winning.
We‟ll say we win (W) if we obtain the $100 at the end and we
lose (L) if we obtain an empty box at the end.
Let„s review the game.
a. Monty presents to us three sealed and labeled boxes.
One contains a $100 bill and the other two contain nothing.
Let‟s say we select box 2 to keep.
c. Monty shows an empty box. Then Monty gives us the option
to switch boxes. We may
I. stay the course and keep the original box.
1 2 3

37. The Monty Hall Problem
Verification of the Monty Hall Fact
We want to verify that changing our original choice in the
middle of the game actually doubled the chance of winning.
We‟ll say we win (W) if we obtain the $100 at the end and we
lose (L) if we obtain an empty box at the end.
Let„s review the game.
a. Monty presents to us three sealed and labeled boxes.
One contains a $100 bill and the other two contain nothing.
Let‟s say we select box 2 to keep.
c. Monty shows an empty box. Then Monty gives us the option
to switch boxes. We may
I. stay the course and keep the original box.
II. switch our box to the unopened box.
1 2 3

38. The Monty Hall Problem
We look at these two options carefully here.

39. The Monty Hall Problem
We look at these two options carefully here.
I. Stay the course

40. The Monty Hall Problem
We look at these two options carefully here.
I. Stay the course
We‟ve the 3 initial possibilities
1
$100 2 3
1 2 3
$100
1 2 3
$100
Stay the course

41. The Monty Hall Problem
We look at these two options carefully here.
I. Stay the course
We‟ve the 3 initial possibilities and we selected box 2.
1
$100 2 3
1 2
$100 3
1 2 3
$100
Stay the course

42. The Monty Hall Problem
We look at these two options carefully here.
I. Stay the course
We‟ve the 3 initial possibilities and we selected box 2.
We start with the case where the $100 is in our box 2.
1
$100 2 3
1 2
$100 3
1 2 3
$100
Stay the course

43. The Monty Hall Problem
We look at these two options carefully here.
I. Stay the course
We‟ve the 3 initial possibilities and we selected box 2.
We start with the case where the $100 is in our box 2.
Monty opens the empty box 1 (or 3),
1
$100 2 3
1 2
$100 3
1 2 3
$100
Stay the course

44. The Monty Hall Problem
We look at these two options carefully here.
I. Stay the course
We‟ve the 3 initial possibilities and we selected box 2.
We start with the case where the $100 is in our box 2.
Monty opens the empty box 1 (or 3),
1
$100 2 3
1 2
$100 3
1 2 3
$100
Stay the course

45. The Monty Hall Problem
We look at these two options carefully here.
I. Stay the course
We‟ve the 3 initial possibilities and we selected box 2.
We start with the case where the $100 is in our box 2.
Monty opens the empty box 1 (or 3), we keep box 2 and win.
1
$100 2 3
1 2
$100 3
1 2 3
$100
Stay the course

46. The Monty Hall Problem
We look at these two options carefully here.
I. Stay the course
We‟ve the 3 initial possibilities and we selected box 2.
We start with the case where the $100 is in our box 2.
Monty opens the empty box 1 (or 3), we keep box 2 and win.
1
$100 2 3
Monty opens box 1 or 3
1 2
$100 3 We keep box 2 W
1 2 3
$100
Stay the course

47. The Monty Hall Problem
We look at these two options carefully here.
I. Stay the course
We‟ve the 3 initial possibilities and we selected box 2.
We start with the case where the $100 is in our box 2.
Monty opens the empty box 1 (or 3), we keep box 2 and win
If the $100 is in box 1
1
$100 2 3
Monty opens box 1 or 3
1 2
$100 3 We keep box 2 W
1 2 3
$100
Stay the course

48. The Monty Hall Problem
We look at these two options carefully here.
I. Stay the course
We‟ve the 3 initial possibilities and we selected box 2.
We start with the case where the $100 is in our box 2.
Monty opens the empty box 1 (or 3), we keep box 2 and win.
If the $100 is in box 1, Monty show us box 3
1
$100 2 3
Monty opens box 1 or 3
1 2
$100 3 We keep box 2 W
1 2 3
$100
Stay the course

49. The Monty Hall Problem
We look at these two options carefully here.
I. Stay the course
We‟ve the 3 initial possibilities and we selected box 2.
We start with the case where the $100 is in our box 2.
Monty opens the empty box 1 (or 3), we keep box 2 and win.
If the $100 is in box 1, Monty show us box 3 and we lose.
1
$100 2 3
Monty opens box 1 or 3
1 2
$100 3 We keep box 2 W
1 2 3
$100
Stay the course

50. The Monty Hall Problem
We look at these two options carefully here.
I. Stay the course
We‟ve the 3 initial possibilities and we selected box 2.
We start with the case where the $100 is in our box 2.
Monty opens the empty box 1 (or 3), we keep box 2 and win.
If the $100 is in box 1, Monty show us box 3 and we lose.
Monty opens box 3
1
$100 2 3 We keep box 2 L
Monty opens box 1 or 3
1 2
$100 3 We keep box 2 W
1 2 3
$100
Stay the course

51. The Monty Hall Problem
We look at these two options carefully here.
I. Stay the course
We‟ve the 3 initial possibilities and we selected box 2.
We start with the case where the $100 is in our box 2.
Monty opens the empty box 1 (or 3), we keep box 2 and win.
If the $100 is in box 1, Monty show us box 3 and we lose.
Likewise we lose if the $100 is in box 3.
Monty opens box 3
1
$100 2 3 We keep box 2 L
Monty opens box 1 or 3
1 2
$100 3 We keep box 2 W
1 2 3
$100
Stay the course

52. The Monty Hall Problem
We look at these two options carefully here.
I. Stay the course
We‟ve the 3 initial possibilities and we selected box 2.
We start with the case where the $100 is in our box 2.
Monty opens the empty box 1 (or 3), we keep box 2 and win.
If the $100 is in box 1, Monty show us box 3 and we lose.
Likewise we lose if the $100 is in box 3.
Monty opens box 3
1
$100 2 3 We keep box 2 L
Monty opens box 1 or 3
1 2
$100 3 We keep box 2 W
Monty opens box 1
1 2 3
$100 We keep box 2
Stay the course

53. The Monty Hall Problem
We look at these two options carefully here.
I. Stay the course
We‟ve the 3 initial possibilities and we selected box 2.
We start with the case where the $100 is in our box 2.
Monty opens the empty box 1 (or 3), we keep box 2 and win.
If the $100 is in box 1, Monty show us box 3 and we lose.
Likewise we lose if the $100 is in box 3.
Monty opens box 3
1
$100 2 3 We keep box 2 L
Monty opens box 1 or 3
1 2
$100 3 We keep box 2 W
Monty opens box 1
1 2 3
$100 We keep box 2 L
Stay the course

54. The Monty Hall Problem
We look at these two options carefully here.
I. Stay the course
We‟ve the 3 initial possibilities and we selected box 2.
We start with the case where the $100 is in our box 2.
Monty opens the empty box 1 (or 3), we keep box 2 and win.
If the $100 is in box 1, Monty show us box 3 and we lose.
Likewise we lose if the $100 is in box 3.
Monty opens box 3
1
$100 2 3 We keep box 2 L
Monty opens box 1 or 3
1 2
$100 3 We keep box 2 P(W) = 1/3
W
Monty opens box 1
1 2 3
$100 We keep box 2 L
Stay the course

55. The Monty Hall Problem
We look at these two options carefully here.
I. Stay the course
We‟ve the 3 initial possibilities and we selected box 2.
We start with the case where the $100 is in our box 2.
Monty opens the empty box 1 (or 3), we keep box 2 and win.
If the $100 is in box 1, Monty show us box 3 and we lose.
Likewise we lose if the $100 is in box 3. The revelation of the
empty box is treated as a nonevent, not warranted for action.
Monty opens box 3
1
$100 2 3 We keep box 2 L
Monty opens box 1 or 3
1 2
$100 3 We keep box 2 P(W) = 1/3
W
Monty opens box 3
1 2 3
$100 We keep box 2 L
Stay the course

56. The Monty Hall Problem
II. Switch our 1st choice with the unopened box
We list the 3 possibilities once more.
1
$100 2 3
1 2
$100 3
1 2 3
$100
Switch the box

57. The Monty Hall Problem
II. Switch our 1st choice with the unopened box
We list the 3 possibilities once more.
We start with the case where the $100 is in our box 2.
1
$100 2 3
1 2
$100 3
1 2 3
$100
Switch the box

58. The Monty Hall Problem
II. Switch our 1st choice with the unopened box
We list the 3 possibilities once more.
We start with the case where the $100 is in our box 2. Monty
opens box 1 (or 3),
1
$100 2 3
1 2
$100 3
1 2 3
$100
Switch the box

59. The Monty Hall Problem
II. Switch our 1st choice with the unopened box
We list the 3 possibilities once more.
We start with the case where the $100 is in our box 2. Monty
opens box 1 (or 3), we switch to box 3 (or 1) and lose.
1
$100 2 3
1 2
$100 3
1 2 3
$100
Switch the box

60. The Monty Hall Problem
II. Switch our 1st choice with the unopened box
We list the 3 possibilities once more.
We start with the case where the $100 is in our box 2. Monty
opens box 1 (or 3), we switch to box 3 (or 1) and lose.
1
$100 2 3
Monty opens box 1
1 2
$100 3 We switch to box 3 L
1 2 3
$100
Switch the box

61. The Monty Hall Problem
II. Switch our 1st choice with the unopened box
We list the 3 possibilities once more.
We start with the case where the $100 is in our box 2. Monty
opens box 1 (or 3), we switch to box 3 (or 1) and lose.
If the $100 is in box 1,
1
$100 2 3
Monty opens box 1
1 2
$100 3 We switch to box 3 L
1 2 3
$100
Switch the box

62. The Monty Hall Problem
II. Switch our 1st choice with the unopened box
We list the 3 possibilities once more.
We start with the case where the $100 is in our box 2. Monty
opens box 1 (or 3), we switch to box 3 (or 1) and lose.
If the $100 is in box 1, Monty shows us box 3
1
$100 2 3
Monty opens box 1
1 2
$100 3 We switch to box 3 L
1 2 3
$100
Switch the box

63. The Monty Hall Problem
II. Switch our 1st choice with the unopened box
We list the 3 possibilities once more.
We start with the case where the $100 is in our box 2. Monty
opens box 1 (or 3), we switch to box 3 (or 1) and lose.
If the $100 is in box 1, Monty shows us box 3, we switch to
box 1
1
$100 2 3
Monty opens box 1
1 2
$100 3 We switch to box 3 L
1 2 3
$100
Switch the box

64. The Monty Hall Problem
II. Switch our 1st choice with the unopened box
We list the 3 possibilities once more.
We start with the case where the $100 is in our box 2. Monty
opens box 1 (or 3), we switch to box 3 (or 1) and lose.
If the $100 is in box 1, Monty shows us box 3, we switch to
box 1 and win.
1
$100 2 3
Monty opens box 1
1 2
$100 3 We switch to box 3 L
1 2 3
$100
Switch the box

65. The Monty Hall Problem
II. Switch our 1st choice with the unopened box
We list the 3 possibilities once more.
We start with the case where the $100 is in our box 2. Monty
opens box 1 (or 3), we switch to box 3 (or 1) and lose.
If the $100 is in box 1, Monty shows us box 3, we switch to
box 1 and win.
Monty opens box 3
1
$100 2 3 We switch to box 1 W
Monty opens box 1
1 2
$100 3 We switch to box 3 L
1 2 3
$100
Switch the box

66. The Monty Hall Problem
II. Switch our 1st choice with the unopened box
We list the 3 possibilities once more.
We start with the case where the $100 is in our box 2. Monty
opens box 1 (or 3), we switch to box 3 (or 1) and lose.
If the $100 is in box 1, Monty shows us box 3, we switch to
box 1 and win. Similarly if the $100 is in box 3,
Monty opens box 3
1
$100 2 3 We switch to box 1 W
Monty opens box 1
1 2
$100 3 We switch to box 3 L
1 2 3
$100
Switch the box

67. The Monty Hall Problem
II. Switch our 1st choice with the unopened box
We list the 3 possibilities once more.
We start with the case where the $100 is in our box 2. Monty
opens box 1 (or 3), we switch to box 3 (or 1) and lose.
If the $100 is in box 1, Monty shows us box 3, we switch to
box 1 and win. Similarly if the $100 is in box 3, Monty opens
box 1,
Monty opens box 3
1
$100 2 3 We switch to box 1 W
Monty opens box 1
1 2
$100 3 We switch to box 3 L
1 2 3
$100
Switch the box

68. The Monty Hall Problem
II. Switch our 1st choice with the unopened box
We list the 3 possibilities once more.
We start with the case where the $100 is in our box 2. Monty
opens box 1 (or 3), we switch to box 3 (or 1) and lose.
If the $100 is in box 1, Monty shows us box 3, we switch to
box 1 and win. Similarly if the $100 is in box 3, Monty opens
box 1, we switch to box 3,
Monty opens box 3
1
$100 2 3 We switch to box 1 W
Monty opens box 1
1 2
$100 3 We switch to box 3 L
1 2 3
$100
Switch the box

69. The Monty Hall Problem
II. Switch our 1st choice with the unopened box
We list the 3 possibilities once more.
We start with the case where the $100 is in our box 2. Monty
opens box 1 (or 3), we switch to box 3 (or 1) and lose.
If the $100 is in box 1, Monty shows us box 3, we switch to
box 1 and win. Similarly if the $100 is in box 3, Monty opens
box 1, we switch to box 3 and win.
Monty opens box 3
1
$100 2 3 We switch to box 1 W
Monty opens box 1
1 2
$100 3 We switch to box 3 L
Monty opens box 1
1 2 3
$100 We switch to box 3 W
Switch the box

70. The Monty Hall Problem
II. Switch our 1st choice with the unopened box
We list the 3 possibilities once more.
We start with the case where the $100 is in our box 2. Monty
opens box 1 (or 3), we switch to box 3 (or 1) and lose.
If the $100 is in box 1, Monty shows us box 3, we switch to
box 1 and win. Similarly if the $100 is in box 3, Monty opens
box 1, we switch to box 3 and win.
Monty opens box 3
1
$100 2 3 We switch to box 1 W
Monty opens box 1
1 2 3 We switch to box 3 L P(W) = 2/3
$100
Monty opens box 1
1 2 3
$100 We switch to box 3 W
Switch the box

71. The Monty Hall Problem
II. Switch our 1st choice with the unopened box
We list the 3 possibilities once more.
We start with the case where the $100 is in our box 2. Monty
opens empty box 1 (or 3), we switch to box 3 and lose.
If the $100 is in box 1, Monty shows us box 3, we switch to
box 1 and win. Similarly if the $100 is in box 3, Monty opens
box 1, we switch to box 3 and win. The revelation of the
empty box caused us to act and switch from losing to winning.
Monty opens box 3
1
$100 2 3 We switch to box 1 W
Monty opens box 1
1 2 3 We switch to box 3 L P(W) = 2/3
$100
Monty opens box 1
1 2 3
$100 We switch to box 3 W
Switch the box

72. The Monty Hall Problem
The Misconception

73. The Monty Hall Problem
The Misconception
Now let‟s look closely at the misconception that there is no
reason to switch.
1 2 3

74. The Monty Hall Problem
The Misconception
Now let‟s look closely at the misconception that there is no
reason to switch.
We reasoned that box 2 and box 3 each has ½ chance of
containing the $100, so of course it makes no difference
whether we trade or not.
1 2 3

75. The Monty Hall Problem
The Misconception
Now let‟s look closely at the misconception that there is no
reason to switch.
We reasoned that box 2 and box 3 each has ½ chance of
containing the $100, so of course it makes no difference
whether we trade or not.
1 2
100 3

76. The Monty Hall Problem
The Misconception
Now let‟s look closely at the misconception that there is no
reason to switch.
We reasoned that box 2 and box 3 each has ½ chance of
containing the $100, so of course it makes no difference
whether we trade or not.
1 2 3
100

77. The Monty Hall Problem
The Misconception
Now let‟s look closely at the misconception that there is no
reason to switch.
We reasoned that box 2 and box 3 each has ½ chance of
containing the $100, so of course it makes no difference
whether we trade or not.
1 2
100 3

78. The Monty Hall Problem
The Misconception
Now let‟s look closely at the misconception that there is no
reason to switch.
We reasoned that box 2 and box 3 each has ½ chance of
containing the $100, so of course it makes no difference
whether we trade or not.
1 2 3
100

79. The Monty Hall Problem
The Misconception
Now let‟s look closely at the misconception that there is no
reason to switch.
We reasoned that box 2 and box 3 each has ½ chance of
containing the $100, so of course it makes no difference
whether we trade or not.
This thinking is faulty because it doesn‟t modeled the game
correctly.
1 2 3

80. The Monty Hall Problem
The Misconception
Now let‟s look closely at the misconception that there is no
reason to switch.
We reasoned that box 2 and box 3 each has ½ chance of
containing the $100, so of course it makes no difference
whether we trade or not.
This thinking is faulty because it doesn‟t modeled the game
correctly. The above reasoning is corrected if the $100 is
inserted randomly into the remaining two boxes after a
boxed is open.
1 2 3

81. The Monty Hall Problem
The Misconception
Now let‟s look closely at the misconception that there is no
reason to switch.
We reasoned that box 2 and box 3 each has ½ chance of
containing the $100, so of course it makes no difference
whether we trade or not.
This thinking is faulty because it doesn‟t modeled the game
correctly. The above reasoning is corrected if the $100 is
inserted randomly into the remaining two boxes after a
boxed is open.
1 2 3
100

82. The Monty Hall Problem
The Misconception
Now let‟s look closely at the misconception that there is no
reason to switch.
We reasoned that box 2 and box 3 each has ½ chance of
containing the $100, so of course it makes no difference
whether we trade or not.
This thinking is faulty because it doesn‟t modeled the game
correctly. The above reasoning is corrected if the $100 is
inserted randomly into the remaining two boxes after a
boxed is open.
1 2
100 3

83. The Monty Hall Problem
The Misconception
Now let‟s look closely at the misconception that there is no
reason to switch.
We reasoned that box 2 and box 3 each has ½ chance of
containing the $100, so of course it makes no difference
whether we trade or not.
This thinking is faulty because it doesn‟t modeled the game
correctly. The above reasoning is corrected if the $100 is
inserted randomly into the remaining two boxes after a
boxed is open.
1 2 3
100

84. The Monty Hall Problem
The Misconception
Now let‟s look closely at the misconception that there is no
reason to switch.
We reasoned that box 2 and box 3 each has ½ chance of
containing the $100, so of course it makes no difference
whether we trade or not.
This thinking is faulty because it doesn‟t modeled the game
correctly. The above reasoning is corrected if the $100 is
inserted randomly into the remaining two boxes after a
boxed is open.
In Monty’s game, the $100 is inserted in the boxes before
the revelation.
1 2 3

85. The Monty Hall Problem
The Misconception
Now let‟s look closely at the misconception that there is no
reason to switch.
We reasoned that box 2 and box 3 each has ½ chance of
containing the $100, so of course it makes no difference
whether we trade or not.
This thinking is faulty because it doesn‟t modeled the game
correctly. The above reasoning is corrected if the $100 is
inserted randomly into the remaining two boxes after a
boxed is open.
In Monty’s game, the $100 is inserted in the boxes before
the revelation. The conditions were preset before the
game, not during the game.
1 2 3

86. The Monty Hall Problem
We‟ll exam the fallacy closely – assuming we selected box 2.

87. The Monty Hall Problem
We‟ll exam the fallacy closely – assuming we selected box 2.
If Monty revealed box 1 as empty, we picture the following.
1 2
100 3
1 2 3
100

88. The Monty Hall Problem
We‟ll exam the fallacy closely – assuming we selected box 2.
If Monty revealed box 1 as empty, we picture the following.
1 2
100 3
1 2 3
100
If Monty revealed box 3 as empty, we picture the following.
1 2
100 3
1
100 2 3

89. The Monty Hall Problem
We‟ll exam the fallacy closely – assuming we selected box 2.
If Monty revealed box 1 as empty, we picture the following.
1 2
100 3
1 2 3
100
If Monty revealed box 3 as empty, we picture the following.
1 2
100 3
1
100 2 3
In both cases, we conclude incorrectly that the probabilities
are ½ vs. ½ that we win or lose.

90. The Monty Hall Problem
But the correct pictures for the game are not static.

91. The Monty Hall Problem
But the correct pictures for the game are not static.
The game must be tracked from the start when the content of
the boxes are determined.

92. The Monty Hall Problem
But the correct pictures for the game are not static.
The game must be tracked from the start when the content of
the boxes are determined.
When the game starts, the possibilities are shown here.
1
$100 2 3
1 2
$100
3
1 2 3
$100

93. The Monty Hall Problem
But the correct pictures for the game are not static.
The game must be tracked from the start when the content of
the boxes are determined.
When the game starts, the possibilities are shown here.
1
$100 2 3
1 2
$100
3
1 2 3
$100
We selected box 2

94. The Monty Hall Problem
But the correct pictures for the game are not static.
The game must be tracked from the start when the content of
the boxes are determined.
When the game starts, the possibilities are shown here.
1
$100 2 3
1 2
$100
3
1 2 3
$100
We selected box 2 and Monty opened an empty box.

95. The Monty Hall Problem
But the correct pictures for the game are not static.
The game must be tracked from the start when the content of
the boxes are determined.
When the game starts, the possibilities are shown here.
1
$100 2 3
1 2
$100
3
1 2 3
$100
We selected box 2 and Monty opened an empty box.
Next set of picture should be the corresponding outcomes
after Monty showed us an empty box.

96. The Monty Hall Problem
1
$100 2 3
1 2
$100
3
1 2 3
$100

97. The Monty Hall Problem
1
$100 2 3 1
$100 2 3
1 2
$100
3
1 2 3
$100
Monty opens a box.

98. The Monty Hall Problem
1
$100 2 3 1
$100 2 3
or
1 2
$100
3 1 2 3
1 2 3
$100
Monty opens a box.

99. The Monty Hall Problem
1
$100 2 3 1
$100 2 3
or
1 2
$100
3 1 2 3
1 2 3
$100 1 2 3
$100
Monty opens a box.

100. The Monty Hall Problem
1
$100 2 3 1
$100 2 3
or
1 2
$100
3 1 2 3
1 2 3
$100 1 2 3
$100
We note that of the three possibilities, most likely we're
holding the wrong box after to the revelation.

101. The Monty Hall Problem
1
$100 2 3 1
$100 2 3
or
1 2
$100
3 1 2 3
1 2 3
$100 1 2 3
$100
We note that of the three possibilities, most likely we're
holding the wrong box after to the revelation. The revelation
increases the probability that the other unopened box is the
winner.

102. The Monty Hall Problem
1
$100 2 3 1
$100 2 3
or
1 2
$100
3 1 2 3
1 2 3
$100 1 2 3
$100
We note that of the three possibilities, most likely we're
holding the wrong box after to the revelation. The revelation
increases the probability that the other unopened box is the
winner.
The switch fails only if my original selection was in the money.
which has probability of 1/3.

103. The Monty Hall Problem
1
$100 2 3 1
$100 2 3
or
1 2
$100
3 1 2 3
1 2 3
$100 1 2 3
$100
We note that of the three possibilities, most likely we're
holding the wrong box after to the revelation. The revelation
increases the probability that the other unopened box is the
winner.
The switch fails only if my original selection was in the money.
which has probability of 1/3.
Hence by switching we get the probability of winning to be 2/3.

104. The Monty Hall Problem
1
$100 2 3 1
$100 2 3
or
1 2
$100
3 1 2 3
1 2 3
$100 1 2 3
$100
We note that of the three possibilities, most likely we're
holding the wrong box after to the revelation. The revelation
increases the probability that the other unopened box is the
winner.
The switch fails only if my original selection was in the money.
which has probability of 1/3.
Hence by switching we get the probability of winning to be 2/3.
Finally let‟s exam the more bizarre implication of the Monty
Hall problem.

105. The Monty Hall Problem
A Question
We are at the intersection of three trails leading to the top of
the mountain. We know only one is passable and the other two
are blocked but we don‟t know which is which. We randomly
selected a trail, say, trail 2 to head for the top. Just after the
selection, Joe came down trail 1 and tells us that trail 1 is
blocked. According to Monty Hall, should we then switch our
choice to trail 3?