3. The Monty Hall problem got its name from the TV game show
Let’s Make a Deal hosted by Monty Hall.
The Monty Hall Problem
4. 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.
The Monty Hall Problem
5. 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.
The Monty Hall Problem
21 3
a. Monty presents to us three sealed and labeled boxes.
6. 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.
The Monty Hall Problem
21 3
a. Monty presents to us three sealed and labeled boxes.
One contains a $100 bill and the other two contain nothing.
7. 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.
The Monty Hall Problem
21 3
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.
8. 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.
The Monty Hall Problem
21 3
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
9. 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.
The Monty Hall Problem
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.
21 3
selected box
One of these or maybe both are empty.
10. 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.
The Monty Hall Problem
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.
21 3
selected box
One of these or maybe both are empty.
11. 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.
The Monty Hall Problem
21 3
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.*
12. 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.
The Monty Hall Problem
21 3
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 saying that in this
instance, let’s assumes that Monty opens box1.
13. 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.
The Monty Hall Problem
21 3
d. Now Monty asks us “Do you want to trade your box with the
remaining one (in this case box 3)*?”.
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.
14. 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.
The Monty Hall Problem
21 3
d. Now Monty asks us “Do you want to trade your box with the
remaining one (in this case box 3)*?”.
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.
*If he had opened box 3 instead,
then of course he would ask us
if we want take box 1 instead.
15. 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.
The Monty Hall Problem
21 3
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,
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.
*If he had opened box 3 instead,
then of course he would ask us
if we want take box 1 instead.
16. 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.
The Monty Hall Problem
21 3
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?
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.
17. To most people, this is a bizarre proposition
The Monty Hall Problem
21 3
18. To most people, this is a bizarre proposition
The Monty Hall Problem
21 3100
Its here.
19. To most people, this is a bizarre proposition
The Monty Hall Problem
21 3100
or here.
20. To most people, this is a bizarre proposition
The Monty Hall Problem
21 3100
here
21. To most people, this is a bizarre proposition
The Monty Hall Problem
21 3100
or here.
22. 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.
The Monty Hall Problem
21 3100
or here.
23. 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.
The Monty Hall Problem
21 3
Surprisingly the right action is to trade away our 1st choice.
24. 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.
The Monty Hall Problem
21 3
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.
25. 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.
The Monty Hall Problem
21 3
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
• address some of its implications
26. The Monty Hall Problem
Verification of the Monty Hall Fact
27. Verification of the Monty Hall Fact
The Monty Hall Problem
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
21 3
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.
31. The Monty Hall Problem
21 3
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.
32. The Monty Hall Problem
21 3
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.
33. The Monty Hall Problem
1 3
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.
2
34. The Monty Hall Problem
1 3
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.
2
c. Monty shows an empty box.
35. The Monty Hall Problem
1 3
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.
2
c. Monty shows an empty box. Then Monty gives us the option
to switch boxes.
36. The Monty Hall Problem
1 3
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.
2
I. stay the course and keep the original box.
c. Monty shows an empty box. Then Monty gives us the option
to switch boxes. We may
37. The Monty Hall Problem
1 3
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.
2
I. stay the course and keep the original box.
II. switch our box to the unopened box.
c. Monty shows an empty box. Then Monty gives us the option
to switch boxes. We may
38. The Monty Hall Problem
We look at these two options carefully here.
39. The Monty Hall Problem
I. Stay the course
We look at these two options carefully here.
40. The Monty Hall Problem
I. Stay the course
We’ve the 3 initial possibilities
Stay the course
We look at these two options carefully here.
1 3
1 3
1 3
$100
$100
2
$100
2
2
41. The Monty Hall Problem
I. Stay the course
We’ve the 3 initial possibilities and we selected box 2.
Stay the course
We look at these two options carefully here.
21 3
21 3
21 3
$100
$100
$100
42. The Monty Hall Problem
I. Stay the course
21 3
21 3
21 3
$100
$100
$100
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.
We look at these two options carefully here.
43. The Monty Hall Problem
We look at these two options carefully here.
I. Stay the course
21 3
21 3
21 3
$100
$100
$100
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),
44. The Monty Hall Problem
We look at these two options carefully here.
I. Stay the course
21 3
21 3
21 3
$100
$100
$100
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),
45. The Monty Hall Problem
We look at these two options carefully here.
I. Stay the course
21 3
21 3
21 3
$100
$100
$100
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.
46. The Monty Hall Problem
We look at these two options carefully here.
I. Stay the course
21 3
21 3
21 3
$100
$100
$100
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.
W
Monty opens box 1 or 3
We keep box 2
47. The Monty Hall Problem
We look at these two options carefully here.
I. Stay the course
21 3
21 3
21 3
$100
$100
$100
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
W
Monty opens box 1 or 3
We keep box 2
If the $100 is in box 1
48. The Monty Hall Problem
We look at these two options carefully here.
I. Stay the course
21 3
21 3
21 3
$100
$100
$100
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.
W
Monty opens box 1 or 3
We keep box 2
If the $100 is in box 1, Monty show us box 3
49. The Monty Hall Problem
We look at these two options carefully here.
I. Stay the course
21 3
21 3
21 3
$100
$100
$100
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.
W
Monty opens box 1 or 3
We keep box 2
If the $100 is in box 1, Monty show us box 3 and we lose.
50. The Monty Hall Problem
We look at these two options carefully here.
I. Stay the course
21 3
21 3
21 3
$100
$100
$100
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.
W
Monty opens box 1 or 3
We keep box 2
If the $100 is in box 1, Monty show us box 3 and we lose.
L
Monty opens box 3
We keep box 2
51. The Monty Hall Problem
I. Stay the course
21 3
21 3
21 3
$100
$100
$100
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.
W
Monty opens box 1 or 3
We keep box 2
If the $100 is in box 1, Monty show us box 3 and we lose.
L
Monty opens box 3
We keep box 2
Likewise we lose if the $100 is in box 3.
We look at these two options carefully here.
52. The Monty Hall Problem
I. Stay the course
21 3
21 3
21 3
$100
$100
$100
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.
W
Monty opens box 1 or 3
We keep box 2
If the $100 is in box 1, Monty show us box 3 and we lose.
L
Monty opens box 3
We keep box 2
Likewise we lose if the $100 is in box 3.
Monty opens box 1
We keep box 2
We look at these two options carefully here.
53. The Monty Hall Problem
I. Stay the course
21 3
21 3
21 3
$100
$100
$100
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.
W
Monty opens box 1 or 3
We keep box 2
If the $100 is in box 1, Monty show us box 3 and we lose.
L
Monty opens box 3
We keep box 2
Likewise we lose if the $100 is in box 3.
L
Monty opens box 1
We keep box 2
We look at these two options carefully here.
54. The Monty Hall Problem
I. Stay the course
21 3
21 3
21 3
$100
$100
$100
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.
W
Monty opens box 1 or 3
We keep box 2
If the $100 is in box 1, Monty show us box 3 and we lose.
L
Monty opens box 3
We keep box 2
Likewise we lose if the $100 is in box 3.
L
Monty opens box 1
We keep box 2
P(W) = 1/3
We look at these two options carefully here.
55. The Monty Hall Problem
I. Stay the course
21 3
21 3
21 3
$100
$100
$100
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.
W
Monty opens box 1 or 3
We keep box 2
If the $100 is in box 1, Monty show us box 3 and we lose.
L
Monty opens box 3
We keep box 2
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.
L
Monty opens box 3
We keep box 2
P(W) = 1/3
We look at these two options carefully here.
56. The Monty Hall Problem
II. Switch our 1st choice with the unopened box
57. The Monty Hall Problem
21 3
21 3
21 3
$100
$100
$100
Switch the box
II. Switch our 1st choice with the unopened box
We list the 3 possibilities once more.
58. The Monty Hall Problem
21 3
21 3
21 3
$100
$100
$100
Switch the box
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.
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),
21 3
21 3
21 3
$100
$100
$100
Switch the box
60. The Monty Hall Problem
2 3$100
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
21 3$100
Switch the box
21 3$100
61. The Monty Hall Problem
2 3$100 L
Monty opens box 1
We switch to box 3
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
21 3$100
Switch the box
21 3$100
62. The Monty Hall Problem
2 3$100 L
Monty opens box 1
We switch to box 3
II. Switch our 1st choice with the unopened box
1
21 3$100
Switch the box
21 3$100
If the $100 is in box 1,
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.
63. The Monty Hall Problem
2 3$100 L
Monty opens box 1
We switch to box 3
II. Switch our 1st choice with the unopened box
1
21 3$100
Switch the box
21 3$100
If the $100 is in box 1, Monty shows us box 3
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.
64. The Monty Hall Problem
2 3$100 L
Monty opens box 1
We switch to box 3
II. Switch our 1st choice with the unopened box
1
21 3$100
Switch the box
21 3$100
If the $100 is in box 1, Monty shows us box 3, we switch to
box 1
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.
65. The Monty Hall Problem
2 3$100 L
Monty opens box 1
We switch to box 3
II. Switch our 1st choice with the unopened box
1
21 3$100
Switch the box
21 3$100
If the $100 is in box 1, Monty shows us box 3, we switch to
box 1 and win.
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.
66. The Monty Hall Problem
2 3$100 L
Monty opens box 1
We switch to box 3
II. Switch our 1st choice with the unopened box
1
21 3$100
Switch the box
21 3$100
If the $100 is in box 1, Monty shows us box 3, we switch to
box 1 and win.
W
Monty opens box 3
We switch to box 1
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.
67. The Monty Hall Problem
2 3$100 L
Monty opens box 1
We switch to box 3
II. Switch our 1st choice with the unopened box
1
21 3$100
Switch the box
21 3$100
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,
W
Monty opens box 3
We switch to box 1
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.
68. The Monty Hall Problem
2 3$100 L
Monty opens box 1
We switch to box 3
II. Switch our 1st choice with the unopened box
1
21 3$100
Switch the box
21 3$100
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,
W
Monty opens box 3
We switch to box 1
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.
69. The Monty Hall Problem
2 3$100 L
Monty opens box 1
We switch to box 3
II. Switch our 1st choice with the unopened box
1
21 3$100
Switch the box
21 3$100
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,
W
Monty opens box 3
We switch to box 1
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.
70. The Monty Hall Problem
2 3$100 L
Monty opens box 1
We switch to box 3
II. Switch our 1st choice with the unopened box
1
21 3$100
Switch the box
21 3$100
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.
W
Monty opens box 3
We switch to box 1
W
Monty opens box 1
We switch to box 3
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.
71. The Monty Hall Problem
2 3$100 L
Monty opens box 1
We switch to box 3
II. Switch our 1st choice with the unopened box
1
21 3$100
Switch the box
21 3$100
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.
W
Monty opens box 3
We switch to box 1
W
Monty opens box 1
We switch to box 3
P(W) = 2/3
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.
72. The Monty Hall Problem
2 3$100 L
Monty opens box 1
We switch to box 3
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
21 3$100
Switch the box
21 3$100
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.
W
Monty opens box 3
We switch to box 1
W
Monty opens box 1
We switch to box 3
P(W) = 2/3
74. The Monty Hall Problem
21 3
Now let’s look closely at the misconception that there is no
reason to switch.
The Misconception
75. 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.
The Monty Hall Problem
21 3
Now let’s look closely at the misconception that there is no
reason to switch.
The Misconception
76. The Monty Hall Problem
21 3100
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.
Now let’s look closely at the misconception that there is no
reason to switch.
The Misconception
77. The Monty Hall Problem
21 3100
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.
Now let’s look closely at the misconception that there is no
reason to switch.
The Misconception
78. The Monty Hall Problem
21 3100
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.
Now let’s look closely at the misconception that there is no
reason to switch.
The Misconception
79. The Monty Hall Problem
21 3100
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.
Now let’s look closely at the misconception that there is no
reason to switch.
The Misconception
80. 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.
The Monty Hall Problem
1
Now let’s look closely at the misconception that there is no
reason to switch.
The Misconception
2 3
This thinking is faulty because it doesn’t modeled the game
correctly.
81. 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.
The Monty Hall Problem
1
Now let’s look closely at the misconception that there is no
reason to switch.
The Misconception
2 3
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.
82. 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.
The Monty Hall Problem
1
Now let’s look closely at the misconception that there is no
reason to switch.
The Misconception
2 3100
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.
83. 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.
The Monty Hall Problem
1
Now let’s look closely at the misconception that there is no
reason to switch.
The Misconception
2 3
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.
100
84. 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.
The Monty Hall Problem
1
Now let’s look closely at the misconception that there is no
reason to switch.
The Misconception
2 3100
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.
85. 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.
The Monty Hall Problem
Now let’s look closely at the misconception that there is no
reason to switch.
The Misconception
2 3
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
86. 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.
The Monty Hall Problem
Now let’s look closely at the misconception that there is no
reason to switch.
The Misconception
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.
2 31
87. The Monty Hall Problem
We’ll exam the fallacy closely – assuming we selected box 2.
88. The Monty Hall Problem
If Monty revealed box 1 as empty, we picture the following.
1 3
1 1002
2
3
100
We’ll exam the fallacy closely – assuming we selected box 2.
89. The Monty Hall Problem
If Monty revealed box 1 as empty, we picture the following.
1 3
1 1002
2
3
100
We’ll exam the fallacy closely – assuming we selected box 2.
2
2100
1
1100
3
3
If Monty revealed box 3 as empty, we picture the following.
90. The Monty Hall Problem
If Monty revealed box 1 as empty, we picture the following.
1 3
1 1002
2
3
100
We’ll exam the fallacy closely – assuming we selected box 2.
2
2100
1
1100
3
3
In both cases, we conclude incorrectly that the probabilities
are ½ vs. ½ that we win or lose.
If Monty revealed box 3 as empty, we picture the following.
91. The Monty Hall Problem
But the correct pictures for the game are not static.
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.
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 3
1 3
1
$100
3$100
2
2
2
$100
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 3
1 3
1
$100
3$100
We selected box 2
2
2
2
$100
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 3
1 3
1
$100
3$100
We selected box 2 and Monty opened an empty box.
2
2
2
$100
96. 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 3
1 3
1
$100
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.
2
2
2
$100
98. The Monty Hall Problem
21 3
21 3
21
$100
3
$100
$100
21 3$100
Monty opens a box.
99. The Monty Hall Problem
21 3
21 3
21
$100
3
$100
$100
21 3$100
2 31
or
Monty opens a box.
100. The Monty Hall Problem
21 3
21 3
21
$100
3
$100
$100
21 3
21
$100
3$100
2 31
or
Monty opens a box.
101. The Monty Hall Problem
We note that of the three possibilities, most likely we're
holding the wrong box after to the revelation.
21 3
21 3
21
$100
3
$100
$100
21 3
21
$100
3$100
2 31
or
102. The Monty Hall Problem
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.
21 3
21 3
21
$100
3
$100
$100
21 3
21
$100
3$100
2 31
or
103. The Monty Hall Problem
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.
21 3
21 3
21
$100
3
$100
$100
21 3
21
$100
3$100
2 31
or
104. The Monty Hall Problem
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.
21 3
21 3
21
$100
3
$100
$100
21 3
21
$100
3$100
2 31
or
105. The Monty Hall Problem
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.
21 3
21 3
21
$100
3
$100
$100
21 3
21
$100
3$100
2 31
or
106. 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.
The Monty Hall Problem
We randomly select a trail, say, trail 2. Just after the selection,
Joe came down trail 1 and tells us that trail 1 is blocked.
Should then switch our choice to trail 3?
An Interesting Question
