MontyHall MontyHall Conditional Conditional Probability - - PowerPoint PPT Presentation

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MontyHall MontyHall Conditional Conditional Probability - - PowerPoint PPT Presentation

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MathematicsforComputerScience MathematicsforComputerScience MIT 6.042J/18.062J MIT 6.042J/18.062J MontyHall MontyHall Conditional Conditional Probability Probability oftenconfusing AlbertRMeyer, May3,2013 AlbertRMeyer,

slide-1
SLIDE 1

Albert R Meyer, May 3, 2013 condmonty.1

Monty Hall Conditional Probability

Mathematics for Computer Science

MIT 6.042J/18.062J

condmonty.1 Albert R Meyer, May 3, 2013 condmonty.2

Monty Hall Conditional Probability

  • ften confusing

Mathematics for Computer Science

MIT 6.042J/18.062J

condmonty.2 Albert R Meyer, May 3, 2013 1/3 Prize location Door Picked Door Opened 1 1 2 3 2 2 1 1 3 3 3 2 2 2 3 3 2 3 3 1 1 2 1 1

1/3 1/3 1/3

1/3 1/3 1/3

1/3 1/3 1/3 1/3 1/3

1/2 1/2

1/2 1/2 1/2 1/2

1

1/18 1/18 1/9 1/9 1/9 1/18 1/18 1/9 1/9 1/9 1/18 1/18

1

1 1 1 1

condmonty.3

(1,1,2) (1,1,3) Monty Hall Probabilities (1,2,3) (1,3,2) (3,1,1) (3,2,1) (3,3,2) (3,3,1)

goat at 2

         

Albert R Meyer, May 3, 2013 1/3 Prize location Door Picked Door Opened 1 1 2 3 2 2 1 1 3 3 3 2 2 2 3 3 2 3 3 1 1 2 1 1

1/3 1/3 1/3

1/3 1/3 1/3

1/3 1/3 1/3 1/3 1/3

1/2 1/2

1/2 1/2 1/2 1/2

1

1/18 1/18 1/9 1/9 1/9 1/18 1/18 1/9 1/9 1/9 1/18 1/18

1

1 1 1 1

condmonty.4

(1,1,2) (1,1,3) Monty Hall Probabilities (1,2,3) (1,3,2)

prize at 1

(3,1,1) (3,2,1) (3,3,2) (3,3,1)

goat at 2

              

1

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SLIDE 2

Albert R Meyer, May 3, 2013

Pr[ prize at 1 | goat at 2]

condmonty.5

Conditional Probability: Monty Hall

Really!

condmonty.5

= 1 2

Albert R Meyer, May 3, 2013

    

1/3 Prize location Door Picked Door Opened 1 1 2 3 2 2 1 1 3 3 3 2 2 2 3 3 2 3 3 1 1 2 1 1

1/3 1/3 1/3

1/3 1/3 1/3

1/3 1/3 1/3 1/3 1/3

1/2 1/2

1/2 1/2 1/2 1/2

1

1/18 1/18 1/9 1/9 1/9 1/18 1/18 1/9 1/9 1/9 1/18 1/18

1

1 1 1 1

condmonty.8

(1,1,2) (1,1,3) Monty Hall Probabilities (1,2,3) (1,3,2)

prize at 1

Albert R Meyer, May 3, 2013

    

1/3 Prize location Door Picked Door Opened 1 1 2 3 2 2 1 1 3 3 3 2 2 2 3 3 2 3 3 1 1 2 1 1

1/3 1/3 1/3

1/3 1/3 1/3

1/3 1/3 1/3 1/3 1/3

1/2 1/2

1/2 1/2 1/2 1/2

1

1/18 1/18 1/9 1/9 1/9 1/18 1/18 1/9 1/9 1/9 1/18 1/18

1

1 1 1 1

condmonty.9

(1,1,2) (1,1,3) Monty Hall Probabilities (1,2,3) (1,3,2)

prize at 1 Carol

  • pens 2

(3,1,1) (3,3,2)

] ] ] ]

Albert R Meyer, May 3, 2013

Pr[ prize at 1 | Carol opens 2]

condmonty.10

Conditional Probability: Monty Hall

Likewise!

condmonty.10

= 1 2

2

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SLIDE 3

Albert R Meyer, May 3, 2013 1/3 Prize location Door Picked Door Opened 1 1 2 3 2 2 1 1 3 3 3 2 2 2 3 3 2 3 3 1 1 2 1 1

1/3 1/3 1/3

1/3 1/3 1/3

1/3 1/3 1/3 1/3 1/3

1/2 1/2

1/2 1/2 1/2 1/2

1

1/18 1/18 1/9 1/9 1/9 1/18 1/18 1/9 1/9 1/9 1/18 1/18

1

1 1 1 1

condmonty.13

(1,1,2) (1,1,3) (3,1,2) (2,1,3)

picked 1

Monty Hall Probabilities

] ] ] ] goat at 2

         

Albert R Meyer, May 3, 2013 1/3 Prize location Door Picked Door Opened 1 1 2 3 2 2 1 1 3 3 3 2 2 2 3 3 2 3 3 1 1 2 1 1

1/3 1/3 1/3

1/3 1/3 1/3

1/3 1/3 1/3 1/3 1/3

1/2 1/2

1/2 1/2 1/2 1/2

1

1/18 1/18 1/9 1/9 1/9 1/18 1/18 1/9 1/9 1/9 1/18 1/18

1

1 1 1 1

condmonty.14

(1,1,2) (1,1,3) (3,1,2) (2,1,3)

picked 1

Monty Hall Probabilities

] ] ] ] goat at 2

         

&

Albert R Meyer, May 3, 2013 1/3 Prize location Door Picked Door Opened 1 1 2 3 2 2 1 1 3 3 3 2 2 2 3 3 2 3 3 1 1 2 1 1

1/3 1/3 1/3

1/3 1/3 1/3

1/3 1/3 1/3 1/3 1/3

1/2 1/2

1/2 1/2 1/2 1/2

1

1/18 1/18 1/9 1/9 1/9 1/18 1/18 1/9 1/9 1/9 1/18 1/18

1

1 1 1 1

condmonty.15

(1,1,2) (1,1,3) (3,1,2)

picked 1

Monty Hall Probabilities

& goat at 2

Albert R Meyer, May 3, 2013 condmonty.16

Conditional Probability: Monty Hall

Also!

condmonty.16

= 1 2

Pr[ prize at 1 | picked 1 & goat at 2]

3

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SLIDE 4

Stick or Switch?

Seems the contestant may as well stick, since the probability is 1/2 given what he knows when he

  • chooses. Wait! contestant

knows more than what door he picked & where a goat is, he knows what door Carol opened!

Albert R Meyer, May 3, 2013 condmont condmonty.21 .21

Monty Hall Probabilities

1/18 (1,1,2) ] 1/18 (1,1,3) ] 1/9 1/9 1/9 (2,1,3) ] 1/18 1/18

1 1 2 2 1/3 1/2 2 1 3 3 2 2

1/3 1/2

1

3

1/3

3 3 3 1

1/3

1/3 1/3

1/3 1/3 1

1/2

1/2

1 1/3

1 Prize 1

1/9

picked 1

location 3 1 1/3 1 2 1/3

1/9 (3,1,2) ]

2 1 1/3 1

1/9

Door 1/2 2 3

1/18

Picked 1/2 1 Door

1/18

Opened

Conditional Probability: Monty Hall

So until now, we have been conditioning on the wrong events —a common blunder. Using the correct one:

Albert R Meyer, May 3, 2013 condmont condmonty y.22 .22

Monty Hall Probabilities

1/18 (1,1,2)]] 1/18 (1,1,3) ] 1/9 1/9 (2,1,3) ] 1/9 ] opened 2 1/18 1/18

1 1 2 2 1/3 1/2 2 1 3 3 2 2

1/3 1/2

1

3

1/3

3 3 3 1

1/3

1/3 1/3

1/3 1/3 1

1/2

1/2

1 1/3

1 Prize 1

1/9

picked 1

location 1 1/3 1 2 1/3 3

1/9 (3,1,2)]]

2 1 1/3 1

1/9

Door 1/2 2 3

1/18

]

Picked 1/2 1 Door

1/18

Opened Albert R Meyer, May 3, 2013

condmonty.23

Albert R Meyer, May 3, 2013

condmonty.24

4

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SLIDE 5

Albert R Meyer, May 3, 2013 1/3 Prize location Door Picked Door Opened 1 1 2 3 2 2 1 1 3 3 3 2 2 2 3 3 2 3 3 1 1 2 1 1

1/3 1/3 1/3

1/3 1/3 1/3

1/3 1/3 1/3 1/3 1/3

1/2 1/2

1/2 1/2 1/2 1/2

1

1/18 1/18 1/9 1/9 1/9 1/18 1/18 1/9 1/9 1/9 1/18 1/18

1

1 1 1 1

condmonty.25

(1,1,2) (1,1,3) (3,1,2) (2,1,3)

picked 1

Monty Hall Probabilities

] ] ] ]

  • pened 2

] ] ] ] &

Albert R Meyer, May 3, 2013 1/3 Prize location Door Picked Door Opened 1 1 2 3 2 2 1 1 3 3 3 2 2 2 3 3 2 3 3 1 1 2 1 1

1/3 1/3 1/3

1/3 1/3 1/3

1/3 1/3 1/3 1/3 1/3

1/2 1/2

1/2 1/2 1/2 1/2

1

1/18 1/18 1/9 1/9 1/9 1/18 1/18 1/9 1/9 1/9 1/18 1/18

1

1 1 1 1

condmonty.26

(1,1,2) (3,1,2)

picked 1

Monty Hall Probabilities

] ]

  • pened 2

] ] &

Albert R Meyer, May 3, 2013 1/3 Prize location Door Picked Door Opened 1 1 2 3 2 2 1 1 3 3 3 2 2 2 3 3 2 3 3 1 1 2 1 1

1/3 1/3 1/3

1/3 1/3 1/3

1/3 1/3 1/3 1/3 1/3

1/2 1/2

1/2 1/2 1/2 1/2

1

1/18 1/18 1/9 1/9 1/9 1/18 1/18 1/9 1/9 1/9 1/18 1/18

1

1 1 1 1

condmonty.27

(1,1,2) (3,1,2)

picked 1

Monty Hall Probabilities

  • pened 2

&

Albert R Meyer, May 3, 2013

prize at 1

[picked 1 & opened 2] = { (1,1,2),(3,1,2) }

condmonty.29

Conditional Probability: Monty Hall

condmonty.29

5

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SLIDE 6

Albert R Meyer, May 3, 2013

Pr[ prize at 1 | picked 1 &

  • pened 2]

[picked 1 & opened 2] = { (1,1,2),(3,1,2) }

condmonty.30

Conditional Probability: Monty Hall

condmonty.30

Pr=1/18

X

Pr=1/9

X

Albert R Meyer, May 3, 2013

Pr[ prize at 1 | picked 1 &

  • pened 2]

condmonty.31

Conditional Probability: Monty Hall

condmonty.31

= 1 / 18 1 / 18 + 1 / 9

Albert R Meyer, May 3, 2013

Pr[ prize at 1 | picked 1 &

  • pened 2]

condmonty.32 condmonty.32

= 1 / 18 1 / 18 + 1 / 9 = 1 3 = Pr[sticking wins]

Conditional Probability: Monty Hall

Albert R Meyer, May 3, 2013

Pr[ prize at 1 | picked 1 &

  • pened 2]

condmonty.33 condmonty.33

= 1 / 18 1 / 18 + 1 / 9 = 1 3

Stick or Switch?

= Pr[sticking wins]

6

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SLIDE 7

Switch!

By conditioning on everything the contestant knows, we’ve finally confirmed what we learned earlier:

2 Pr[switching wins] = 3

The 4 Step Method

It’s easy to how so many smart people get confused by Monty Hall.

Albert R Meyer, May 3, 2013 condmont condmonty.34 .34 Albert R Meyer, May 3, 2013 condmonty.35

The 4 Step Method The 4 Step Method

It’s easy to how so many smart people get confused by Monty Hall. Finding the right event to condition on can be tricky. It’s easy to how so many smart people get confused by Monty Hall. Finding the right event to condition on can be

  • tricky. The 4 step method

is a good fall back approach.

Albert R Meyer, May 3, 2013 Albert R Meyer, May 3, 2013 condmonty.36 condmonty.37

7

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SLIDE 8

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6.042J / 18.062J Mathematics for Computer Science

Spring 2015 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.