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Opinionated

Opinionated. Lessons. in Statistics. by Bill Press. #3 Monty Hall. Example: The Monty Hall or Let’s Make a Deal Problem. Three doors Car (prize) behind one door You pick a door, but don’t open it yet

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Opinionated

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  1. Opinionated Lessons in Statistics by Bill Press #3 Monty Hall Professor William H. Press, Department of Computer Science, the University of Texas at Austin

  2. Example: The Monty Hall or Let’s Make a Deal Problem • Three doors • Car (prize) behind one door • You pick a door, but don’t open it yet • Monty then opens one of the other doors, always revealing no car (he knows where it is) • You now get to switch doors if you want • Should you? • Most people reason: Two remaining doors were equiprobable before, and nothing has changed. So doesn’t matter whether you switch or not. • Marilyn vos Savant (“highest IQ person in the world”) famously thought otherwise (Parade magazine, 1990) • Monty Hall, the game host, actually understood this (according to a 1991 interview) • he is alive in his 90s • his daughter is Joanna Gleason, who starred in Sondheim’s “Into the Woods” Professor William H. Press, Department of Computer Science, the University of Texas at Austin

  3. 1 2 ( j ) b h d d H P H O i i i 3 1 1 2 3 ¢ c a r / e n o o r = = = i 1 3 6 , ; ; ( ) l k d l b l W i i 2 o g y o u p c o o r r e a e n g 1 1 1 ( j ) , . P H O 3 ¢ / = 2 ( ) 2 3 6 l d l b l W M i 3 t o g o n y o p e n s o o r r e a e n g , . 1 ( ( j j ) ) ( j ) ( ) P P H H O O P O H P H 3 3 0 3 0 ¢ / / = i i i 3 3 “Without loss of generality…” ignorance of Monty’s preferencebetween 1 and 3, so take 1/2 So you should always switch: doubles your chances! Professor William H. Press, Department of Computer Science, the University of Texas at Austin

  4. l ! F V M i i t t t t e r y m p o r a n e x a m p e a s e r . 1 ( ) h \ b b l " \ " F P H i i i i i t t s e p r o r p r o a y o r p r o r = i 3 ( j ) h \ b b l " \ " F P H O i i i i i 3 t t t t s e p o s e r o r p r o a y o r p o s e r o r i ( j ) h \ d f " \ d " F P O H i i i 3 t t s e e v e n c e a c o r o r e v e n c e i d F B i i i t £ a y e s s a y s p o s e r o r / e v e n c e p r o r Monty Hall and the Reverend Bayes Professor William H. Press, Department of Computer Science, the University of Texas at Austin

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