This document presents the famous Monty Hall problem, which is a probability puzzle that most people initially get wrong. It involves a game show with 3 doors, where a prize is behind one door. A contestant picks a door, then the host opens an empty door from the other 2. The contestant is then offered a choice to stick with their initial door or switch to the other unopened door. Most people think the chances are 50/50, but in reality the chance of winning is higher (2/3) if the contestant switches doors due to the information provided by the host eliminating one door. The document uses several examples and explanations to thoroughly illustrate why the probability favors switching doors in this counterintuitive puzzle

1. THE MONTY HALL

2. WELCOME
My name is Ercan Cem

3. I will try to present

4. the most famous*
puzzle of all times.

5. the most famous*
puzzle of all times.
*arguable

6. It is known as
The Monty Hall

7. POPULAR
BECAUSE

8. COUNTER INTUITIVE

9. THE
PUZZLE

10. You are in a TV show.

11. There are three doors.
1 2 3

12. Behind one of them
there's a prize.

13. It is yours if you
guess correctly.

14. The format of
the show is...

15. First, you make a guess.
1 2 3

16. The host opens an empty
door from the other two.
1 2 3

17. The host knows
where the prize is.

18. He offers you
a new option:

19. You can stick with
your initial choice.

20. OR

21. Select the other
(unopened) door.

22. QUESTION

23. WHAT IS
THE RATIONAL
STRATEGY?

24. IS IT...

25. Stick?
1 2 3

26. OR...

27. Select the other?
1 2 3

28. OR...

29. Is this just a
random guess?
1 2 3

30. The usual* answer is
that the final choice is
just a random guess.

31. The usual* answer is
that the final choice is
just a random guess.
*underestimate

32. People tend
to think that

33. at first, each door
has the same one
in three chance.

34. (which is true)

35. So, the chance that
our guess is correct
is one in three.

36. (which is true)

37. Once one of the
doors is opened
1 2 3

38. the unopened
two doors
1 2 3

39. share the chance
of the opened door.

40. THEREFORE

41. The final decision
is a random guess.
1 2 3

42. WRONG!

43. The answer is:

44. If you want to
increase your
chance,

45. you must switch to the
other door.
1 2 3

46. (we exclude the cases)

47. (that you are superstitious
about your inital guess)

48. OR

49. You are a clairvoyant.

50. HERE IS THE
EXPLANATION

51. (without getting too
technical)

52. FIRST TRY

53. When you made
your initial guess,
1 2 3

54. your chance of
being correct was
one in three.

55. ALSO,

56. As a fact,
we know that

57. at least one of
the remaining
doors is empty.

58. AGREE?

59. GOOD.

60. FURTHERMORE,

61. The host will always
have a choice to open
an empty door.

62. SO,

63. when he opens
an empty door,
1 2 3

64. he does not provide
an extra information.

65. He does not provide
anything new.

66. We already know
that at least one
other door is empty.

67. The chance that our
initial guess is correct
is one in three.

68. Nothing changed
since then.

69. After the door
is opened,

70. that chance is
still one in three.

71. HOWEVER

72. When two doors
are left,

73. it cannot be the case
that each has a chance
of one in three.

74. The TOTAL chance
must add up to 1.

75. THEREFORE

76. The chance that the
prize is behind the other
door is two in three.

78. UNLESS

79. You are very supertitious,

80. OR

81. CLAIRVOYANT

82. You must switch.

83. Not convinced?

84. SECOND TRY

85. Imagine that just before
the host opens one of the
remaining doors,

86. the aliens kidnap him.

87. Right at this moment, we
can reason as follows:

88. The host would either open
Door- 2, or Door -3.

89. Say he opened Door -2
1 2 3

90. Then, only Door -1 and
Door-3 would be left.
1 2 3

91. The chance that the
prize is behind Door-1
would be 50%.

92. Instead, say he
opened Door -3
1 2 3

93. Then, only Door -1 and
Door-2 would be left.
1 2 3

94. The chance that the
prize is behind Door-1
would be 50%.

95. Hmmm!

96. Isn't that
reasoning a bit...

97. NAIVE?

98. BECAUSE

99. It suggests that

100. the chance that our initial
guess is correct was
50% in the first place.

101. We KNOW that
that is WRONG!

102. It is
one in three.

103. It cannot increase
all of a sudden.

104. THEREFORE

105. The chance that the
prize is behind the other
door is two in three.

106. Not convinced?

107. LAST TRY

108. LAST TRY
(You’d better be convinced this time.)

110. Imagine a deck of cards.

111. (52 cards that is.)

112. You pick one.

113. If it is the ace of spades,

114. you win.

115. You picked one.

116. 51 cards are left.

117. I KNOW which card
is the ace of spades.

118. Among the
other 51 cards,

119. at least 50 of them is
NOT the ace of spades.

120. I turn 50 cards
upside down.

121. Right now, there are
two unopened cards
on the table...

122. yours, and
the 52nd card.

123. Do you
REALLY
think that

124. the chance that your pick is
the ace of spades is 50%?

125. TIME TO
WRAP UP

126. In the last example,
the essence of the
puzzle is much clearer.

127. That is because we
were dealing with
larger quantities.

128. Things look fuzzier
when dealing with
only three objects.

129. SIDE NOTE

130. In the original problem,
suppose that the host
opens a random door.

131. (Translation: It could be the
case that the opened door
is the one with the prize.)

132. If the opened
door is empty,

133. then the chance that our
initial guess is correct
indeed rises to one in two.

134. HENCE

135. Everything
depends on

136. whether the door was
opened at random,

137. or with the knowledge
that where the prize is.

138. Thanks for
sparing your time.

139. (A simple Google search will get you
to endless sources on Monty Hall.
There’s even a book devoted on it.)

140. (If you enjoyed this puzzle, you can
find more at my blog Mathzzle:
puzzle.ercancem.com.)

141. UNTIL
NEXT
TIME

142. SEE YA!