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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.
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Suppose you pick door number 1.
The host, who knows what's behind the other
doors, opens another door, say door number 3.
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Original selection
Host opens door 3
Door number 3 has a goat behind it. The host
then says to you, 'Do you want to pick door
number 2? Is it to your advantage to switch?"
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If you switched you would have won the car!
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Monty Hall Problem

• From a television show called Lets Make a
  Deal.
• The contestant has a probability of 1/3 of
  choosing the prize if a random choice of doors is
  made.

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Monty Hall Problem

• Let C1be the event that the car is behind door 1.
• Let H1 be the event that the host opens door 1
• and so on for doors 1,2 and 3.
• Suppose you choose door 1.
• Probability of winning a car if you stick to your
  original choice 1/3
• Probability of winning a car if you change
• Probability that the host opens door 3 and the
  car is behind door 2
• or probability that the host opens door 2 and the
  car is behind door 3
• P( H3 n C2 ) or P( H2 n C3 )
• P(C2) P(H3 C2) P(C3) P( H2 C3)
• (1/3) (1) (1/3) (1) 2/3