A Classic Exercise in Probability

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The Monty Hall Problem

• A Classic Exercise in Probability

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Based on the popular game show hosted by Monty
Hall
Lets Make a Deal!
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• Originally posed in the Ask Marylin"
  question-and-answer column of the Parade
  magazine.
  

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The Question

• Suppose you're on a game show, and you're given
  the choice of three doors. Behind one door is a
  car, behind the others, goats. You pick a door,
  say number 1, and the host, who knows what's
  behind the doors, opens another door, say number
  3, which has a goat. He says to you, "Do you want
  to pick door number 2?" Is it to your advantage
  to switch your choice of doors?

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Lets Make a Deal!

• You will be given 3 doors to pick from.
• One of the three doors contains a grand prize.
• The other two doors contain a goat.

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Make your choice!
Door 3
Door 2
Door 1
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Sorry -- The Goat!
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A New Jaguar Convertible!
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Internet Site with Monty Hall Problem with links
to discussions, solutions, and game simulations
http//math.ucsd.edu/crypto/Monty/Montytitle.html
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Strategy in the Monty Hall Game

• Behind one of the doors is a car -- that you
  really, really want!
• If you pick a door at random, what is the initial
  probability that you will get the car?
• 1/3 (1 car, 2 goats -- 3 doors).

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No-Switching Strategy

• You have decided to keep the door you first chose
• Monty has opened another door to reveal a goat.
• What is the probability that you will lose --
  get a goat?
• 2/3 -- nothing has changed.
• What is the probability that you will win -- get
  the car ?
• 1 - 2/3 1/3.

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Switching Strategy

• You decide to switch doors after the goat is
  revealed.
• What is the probability of not getting the car?
• Answer 1/3
• What is the explanation of this counter-intuitive
  result?

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Three Equally Likely Game Scenarios in The
Switching Strategy
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Possible Game 1

• The prize was originally behind the selected door
  1
• The Host then opens one of the other doors
  revealing the goat.
• The Contestant now switches to the remaining
  unopened door
• The outcome is YOU LOSE!

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Possible Game 2

• The prize was originally behind first unselected
  door (2).
• Monty must then open the other unselected door
  ( 3 ) to show goat.
• The contestant now switches to the remaining
  unopened door
• The outcome is YOU WIN!
  

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Possible Game 3
The prize was originally behind second unselected
door (3). Monty must then open the other
unselected door ( 2 ) to show goat. The
contestant now switches to the remaining unopened
door. The outcome is YOU WIN!

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-- Smarter to Switch!

• In two of the the three games you win!
• In only one of the three games you lose.
• With the switching strategy, the probability of
  winning is 2/3.
• In contrast, the no-switch strategy had a
  probability of winning equal to only 1/3!

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Discussion Question

• How would the probabilities change if there were
  four doors three goats and one car?
• If the value of the car if 30,000 and the value
  of the goat is 300, what is your expected
  winnings if you switch?
• What is your expected winnings if you do not
  switch?