Optical Spy
Welcome to the world of optical illusions
  • Opticals
  • Spy Base
  • Spy Blog
  • Escher
    • Escher
    • Escher in Color
  • Galleries
    • David Arthur-Simons Gallery
    • Optrix Gallery
    • John Langdon Gallery
    • Gianni Sarcone Gallery
    • Redmer Hoekstra Gallery
    • Donald Rust Gallery
    • Optical Spy Gallery
    • Tech Art Gallery
    • Animated Art Gallery
    • Body Art Gallery
    • Guido Daniele Gallery
    • Banksy Gallery
    • David Macdonald Gallery
    • Oleg Shupliak Gallery
    • Ben Heine Gallery >
      • Ben Heine Biography
    • Liu Bolin Gallery
    • Erik Johansson Gallery
    • Major Drapkin Gallery
    • High Speed Photography Gallery
    • Tom French Gallery
    • Willard Wigan Gallery
  • List of Illusions
  • Street Art
    • Street Art
    • Evol Gallery
    • Banksy Gallery
  • Contact
  • About
    • Conditions
  • Slide
  • SiteMap
  • Good Stuff
  • Personal Illusion
  • Store
  • Optical Illusions
  • Oleg Shuplyak
  • Oleg Shuplyak Gallery
  • Untitled

The Monty Hall Paradox

1/26/2013

0 Comments

 
Picture
The Monty Hall problem is a probability puzzle loosely based on the American television game show Let's Make a Deal and named after the show's original host, Monty Hall. The problem, also called the Monty Hall paradox, is a veridical paradox because the result appears odd but is demonstrably true. The Monty Hall problem, in its usual interpretation, is mathematically equivalent to the earlier Three Prisoners problem, and both bear some similarity to the much older Bertrand's box paradox.

The problem was originally posed in a letter by Steve Selvin to the American Statistician in 1975. A well-known statement of the problem was published in Marilyn vos Savant's "Ask Marilyn" column in Parade magazine in 1990:

  • 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 No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

Vos Savant's response was that the contestant should always switch to the other door. If the car is initially equally likely to be behind each door, a player who picks Door 1 and doesn't switch has a 1 in 3 chance of winning the car while a player who picks Door 1 and does switch has a 2 in 3 chance. Consequently, contestants who switch double their chances of winning the car.
Many readers refused to believe that switching is beneficial. After the Monty Hall problem appeared in Parade, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine claiming that vos Savant was wrong. (Tierney 1991) Even when given explanations, simulations, and formal mathematical proofs, many people still do not accept that switching is the best strategy.

The Monty Hall problem has attracted academic interest because the result is surprising and the problem is interesting to formulate. Furthermore, variations of the Monty Hall problem are made by changing the implied assumptions, and the variations can have drastically different consequences. For example, if Monty only offered the contestant a chance to switch when the contestant had initially chosen the car, then the contestant should never switch. Variations of the Monty Hall problem are given below.

Extended problem description

Certain aspects of the host's behavior are not specified in Marilyn vos Savant's wording of the problem. For example, it is not clear if the host considers the position of the prize in deciding whether to open a particular door or is required to open a door under all circumstances (Mueser and Granberg 1999). Almost all sources make the additional assumptions that the car is initially equally likely to be behind each door, that the host must open a door showing a goat, and that he must make the offer to switch. Many sources add to this the assumption that the host chooses at random which door to open if both hide goats, often but not always meaning by that, at random with equal probabilities. The resulting set of assumptions gives what is called "the standard problem" by many sources (Barbeau 2000:87). According to Krauss and Wang (2003:10), even if these assumptions are not explicitly stated, people generally assume them to be the case. A fully unambiguous, mathematically explicit version of the standard problem is:

  • Suppose you're on a game show and you're given the choice of three doors [and will win what is behind the chosen door]. Behind one door is a car; behind the others, goats [unwanted booby prizes]. The car and the goats were placed randomly behind the doors before the show. The rules of the game show are as follows: After you have chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it. If both remaining doors have goats behind them, he chooses one [uniformly] at random. After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door. Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you "Do you want to switch to Door Number 2?" Is it to your advantage to change your choice?
  • —Krauss and Wang 2003:10

Solutions

A simple way to demonstrate that a switching strategy really does win two out of three times on the average is to simulate the game with playing cards (Gardner 1959b; vos Savant 1996:8). Three cards from an ordinary deck are used to represent the three doors; one 'special' card such as the Ace of Spades should represent the door with the car, and ordinary cards, such as the two red twos, represent the goat doors.
Picture
The simulation, using the following procedure, can be repeated several times to simulate multiple rounds of the game. One card is dealt face-down at random to the 'player', to represent the door the player picks initially. Then, looking at the remaining two cards, at least one of which must be a red two, the 'host' discards a red two. If the card remaining in the host's hand is the Ace of Spades, this is recorded as a round where the player would have won by switching; if the host is holding a red two, the round is recorded as one where staying would have won.

By the law of large numbers, this experiment is likely to approximate the probability of winning, and running the experiment over enough rounds should not only verify that the player does win by switching two times out of three, but show why. After one card has been dealt to the player, it is already determined whether switching will win the round for the player; and two times out of three the Ace of Spades is in the host's hand.

If this is not convincing, the simulation can be done with the entire deck, dealing one card to the player and keeping the other 51 (Gardner 1959b; Adams 1990). In this variant the Ace of Spades goes to the host 51 times out of 52, and stays with the host no matter how many non-Ace cards are discarded.

Another simulation, suggested by vos Savant, employs the "host" hiding a penny, representing the car, under one of three cups, representing the doors; or hiding a pea under one of three shells.

Vos Savant's solution

The solution presented by vos Savant in Parade shows the three possible arrangements of one car and two goats behind three doors and the result of switching or staying after initially picking Door 1 in each case:
Picture
A player who stays with the initial choice wins in only one out of three of these equally likely possibilities, while a player who switches wins in two out of three. The probability of winning by staying with the initial choice is therefore 1/3, while the probability of winning by switching is 2/3.
0 Comments



Leave a Reply.

    Optical Illusion Website



    Why not subscribe to our Spy Blog

    Subscribe

    RSS Feed


    Picture
    Picture
    Picture

    Archives

    October 2017
    August 2015
    July 2015
    June 2015
    May 2015
    April 2015
    March 2015
    February 2015
    January 2015
    December 2014
    November 2014
    October 2014
    September 2014
    August 2014
    April 2014
    March 2014
    February 2014
    December 2013
    November 2013
    October 2013
    July 2013
    June 2013
    April 2013
    March 2013
    February 2013
    January 2013


    Illusions

    Sitting in the Pool -  Balloon Dog Illusion -  Deer Tree -  Driving through an Earthquake -  A Mind Bending Optical Illusion -  Houdini Illusion -  Can you read this? -  Faces in the Woods -  Self Filtering Swimming Pool -  Is Sylvester on Twitter? -  Pretty Lady -  Flying on the Beach -  Invisible lady -  Blocks and Lines -  Road to nowhere -  Rubik's Triangle -  Perspective -  What Lies Beneath? -  Escaping Flower -  Which Building is in Front of the Other? -  Steelwork Illusion -  Eyeball in the sink -  Leopard Illusion -  Cliff Face -  Yesterdays News Today -    Checkerboard Illusion -  It's a Small World -  Office Illusion -  The Lamp is a Lie -  Checker Board Illusion -  Can You Sleep? -  Green and blue Optical illusion -  Australian Building Optical Illusion -  Not Wavy At All -  The Winners of the 2014 Best Illusion of the Year Contest -  Is Religion an Illusion? -  Doctor Who? -  Which One is Real? -  Mind Your step -  Animal Illusion -  Animal Illusions -  Pig Owl -  Impossible Jenga -  The Lady Is For Turning -  Moving Balls Illusion -  Building on Tilt - Really -  Building on Tilt -  Psychedelic Dylan -  Man Lands on Moon -  Nice Dancing Spot -  Can you see the hiden Animal? -  Moon Illusions -  Coming or Going? -  Big Ben in the Shade -  Stereographic Monsters -  Spiricles -  Toulouse Lautrec Illusion -  Boat Illusion -  Mondrian Slide Puzzle -  Scaffold Illusion -  Chair Position -  Eye - Optical illusion -  Man changes color of rose -  Invisible Chinese Artist -  Magnifier -  Bendy Straws -  Cow Illusions -  Victorian UFO -  Infinite Zoom Illusions -  Movie Quiz -  Archway -  Butterflies -  Dancers on Stage -  Going Down -  Texas leaf cutter -  Cubes -  It's behind you! -  Cricket Illusion -  Ambiguity -  Tech Art -    Alphabet -  Whoops! -  Magic Eye Video -  Marie Celeste -  Beach Art Illusion -  Can You Name This Famous Couple -  Which Way are the Cars Moving? -  Suzi Perry Optical Illusion -  Black Spot Illusion -  Rainbow IllusioN -  Miniature Food Art -  Mobius Wall -  3D Snake Chalk Art -  The Maric Arrow Optical Illusion -  Three Legged Lovely Optical Illusion -  Amelymeloptical illusion -  This Hot Chocolate is Good Enough to Eat -  Alien Illusion -  Body Illusion -  Back of Hand Illusion -  Union Jack Illusion -  Grid Illusion -  Floating Circles Illusion -  Hidden Message -  Bird Glasses -  Houses -  What's wrong with this picture? -  Frozen Illusion -  Body Paint Babes -  Big Girls Illusion -  Funny Illusions -  Indoor Anamorphic art -  Plate Size Optical Illusion -  Which Cigarette is the Biggest? -  Veg Man -  Amazing Paper Illusion with a Twist -  Left or Right Illusion -  Good or Evil? -  How Many Zebras? -  Spinning Disk Illusion -  Amazing Secret Monitor -  Upside Down Couple Illusion -  A Simple Question of Shape -  Hole or Mirror? -  Hiding in the Stones -  Floating Illusion -  Right Angle illusion -  Two Straight or Wavy Line Illusions -  Block Illusion -  Black & White Illusion -  Hole in the Street -  USB -  Animal silhouette? -  Bart Simpson Illusion -  Balls Illusion -  Kissing the Sphynx -  Crossed Lines Optical illusion -  Bridge Line Illusion -  Between the Lines Optical Illusion -  Pencils Optical Illusion -  Twisted Lines Optical Illusion -  Spear Optical Illusion -  Moving Traffic -  House Tilt Illusion -  Boat After Image -  This will Make You Chuckle -  Moving Lines Illusion -  It's an Up and Down World -  3D Animated GIF Illusion -  The McGurk Effect -  Houdini's ghost -  Ring Pull Barrel -  Extra Square Illusion -  Ladder Illusion -  Ape Tree Illusion -  Moving Words Illusion -  Steak Faces -  Winter Olympic Rings -  Marbles and Eyeballs -  Statue of Liberty Illusion -  Eye Color Illusion -  Which Pencil is Longest? -  Kate Russell Illusion -  Star Trek - Back to the Future -  Box Lid Illusion -  Stairs Illusion -  One Direction Illusion -  Line Illusion -  Spinning Head Illusion -  Toblerone's Logo Illusion -  Physics Illusion -  Amazing Video -    Dice Illusion -  Can You Read This? -  White or Black Crosses Illusion -  Alien Tiger Missing Square Illusion -  Three New images Added to Our Liu Bolin Gallery -  SPY LINES -  Little Girl Laughs Her Head Off -  Patrick Hughes Corner Stores -  Sketchy Cube Illusion -  Seattle UFO -  Big Cubes - Little Cubes -  See Through Tree Illusion -  Grid Shade Test -  3D Dotted Grid -  December 31st, 2013 -  Road Markings -  Stephen Fry - Alan Davies Illusion -  Bruegel's Illusion of a Magician -  Glass Floor? -  Tom French -  Merry Christmas Illusion -  These Four Images Are Just One Single Picture -  Star Shade Illusion -    Crossword -  Strange Bandstand -  Which is the Biggest? -  Holding Up the Moon -  What a Lot of Balls! -  Now that is what you call CamoUflage -  How Many Squares -  A Waist of Time -  Animal Face -  Tartan Illusion -  Ring Illusion -  Scary Room -  Hole or Hill? -  Colorado River Illusion -  Richard Branson Optical Illusion -  Slow Ball 2 -  Train Wheels Illusion -  Disappearing Star Illusion -  Moving Balls Illusion -  A Spyral that is not a Spiral -  Graph Illusion -  Stripey Illusion -  Can you find the cat? -  Le Blanc-Seing - Rene Magritte -  Circling Arrows Illusion -  Parrot Illusion -  Balance Beam Illusion -  CatMan -  Pollyanna Woodward Upside Down Illusion -  Famous Painting Quiz -  Snow Face Here! -  Looking Down -  Honda's Illusion Ad and How It's Made -  How to create chocolate out of nothing -  Hidden Animals -  Checker Board Balls Illusion -  Moving Patterns Illusion -  Penn Jillette Optical Illusion -    Clocks Illusion -  Three Movers -  The Mouse and the Lion Illusion -  Semi Circles -  Old Spyral -  Tyre Art -  How Many Faces? -  Incredible Illusions & Science! -  Vintage Car Illusion -  The Ball Switch Illusion -  What's hiding here? -  Jungle Illusion -  The Hand of God -  Color Illusion -  Bouncing or Crossing Balls? -  Sisters in Arms -  Perspective Motion Illusion -  Charles Darwin -  Spinning Wheel -  Don't Drink and Read! -  Where are these places? -  Tyred Tree! -  Moving Monster truck -  Hypnotic Eye -  Diagonals Illusion -  Color Strips -  Paper Face Illusion -  3 Circles Illusion -  No Swimming in this Lake! -  The Magic Hula Hoop Illusion -  Alan Mason Illusion -  Karun Chandhok -  London Eye Bike Illusion -  Frosty Babe -  In or Out Illusion -  Eye Time -  Little Bike Riders Illusion -  Target Illusion -  Erik Johansson -  Can You Spot the Photographs Hiding in Plain Sight? -  Black and White -  Sun Bean -  Clouds or Tsunami? -  Left or Right -  The Melancholic Princess -  Moving Flowers -  More Hidden Faces -  Hidden Faces -  ForeArms -  Two Trains Coming or Going? -  Is this Train Coming or Going? -  Up and Down -  Color Words Challenge -  Shahrukh Khan Illusion -  Orange cross Illusion -  Pyramid or Skylight Illusion -  Rory Cellan-Jones Illusion -  Office Sign Illusion -  Moving Spy -  Historical Illusions -  Shiva by Luke Brown -  Juggling on the Beach -  Assumptions -  London, the Kitchen of England -  Philosophical Question -  Devil in the Sky Illusion -  Looking Around -  Tiger Nuts Illusion -  Bobbyllew Illusion -  Desktop Puzzle -  Desktop Puzzle Answers -  Hidden Words -  Chair Illusion -  The Tree of Life -  A Clockwork Illusion -  Rollercoaster -  Fly Past -  Snakes Illusion -  Which Line is Longer? -  Short Pack - Long Pack Illusion -  A, B or C? -  Kissing Shadows -  Moving Stone -  Hiding in the Pebbles -  Who are these two? -  Umbrella Illusion -  Be Very Careful -  Hold that Plane -  Dolphin Illusion -  Bulge -  Color Puzzle -  Japanese Fighting Bots -  Pen Holder Illusion -  Garage Door Illusion -  Why the Long Face? -  Key Places -  Invisible Spy -  Orange Spin Illusion -  Brain Numbing Illusion -  Banana Illusion -  Violin Island -  Window Illusion -  Moon Illusion -  Blue Diamond Illusion -  Rabbit Cat -  Waves -    Video No1 -  Owl Eyes Illusion -  Halo Halo -  Rock Climber -  London Eye Illusion -  Disappearing Stars -  Barber Shop Illusion -  Barber shop Illusion Revealed -  Find the Stars -  Spinners -  Optical Owl -  LadyBird -  Hollywood Face Illusion -  Strips -  Vortex -  Eyeball -  Amazing Anamorphic Illusions! -  Heavenly Body -  Lovers -  Pick the Pattern -  Lizard Haircut -  Moving Shapes -  Brick Wall Illusion -  Rotating Disk -  Don't Lose Your Head -  Eye Eye -  Magic Eye Message Illusion -  Mona's Pets -  
Copyright © opticalspy.com. All rights reserved. | News