The Case of Sleeping Beauty

Tyler Seacrest

April 29, 2009

Bayes’ Rule Example

Bayes’ Rule Example

Bayes’ Rule Example

50%

50%

Bayes’ Rule Example

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Bayes’ Rule Example

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Question: What is the probability a black ball was put in?

Bayes’ Rule Example

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Bayes’ Rule Example

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Bayes’ Rule Example

? Momentof truth!

Bayes’ Rule Example

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Case 1

Bayes’ Rule Example

Case 1

(Probability a black ball was initially put inis 100%)

Bayes’ Rule Example

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Bayes’ Rule Example

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Case 2

Bayes’ Rule Example

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Case 2

(Probability a black ball was put in goes down)

Bayes’ Rule Example

Case 2

Possible contents of the urn

Bayes’ Rule Example

Case 2

Possible contents of the urn

I One of these three balls were choosen, and each case isequally likely

I Hence, the probability a black ball was put in initially drops to1/3.

Bayes’ Rule Example

Case 2

Possible contents of the urn

I One of these three balls were choosen, and each case isequally likely

I Hence, the probability a black ball was put in initially drops to1/3.

The Case of Sleeping Beauty

I A subject known as SB takes part in an experiment at theACME probability labs. She has full knowledge of all details ofthe experiment.

I Sunday night, SB goes to bed. While she is asleep, a coin isflipped.

I She is awoken Monday morning, and asked, “From your pointof view, what is the probability the coin came up heads?”. Ifthe coin actually did come up heads, this is the end of theexperiment.

I If the coin came up tails, SB is injected with a special drugcocktail that will put her back to sleep and make hercompletely forget Monday’s events.

I She is then awoken on Tuesday morning (which, to her, seemslike Monday morning) and asked “From your point of view,what is the probability the coin came up heads?”

I If you were in SB’s shoes, how would you answer?

The Case of Sleeping Beauty

I A subject known as SB takes part in an experiment at theACME probability labs. She has full knowledge of all details ofthe experiment.

I Sunday night, SB goes to bed. While she is asleep, a coin isflipped.

I She is awoken Monday morning, and asked, “From your pointof view, what is the probability the coin came up heads?”. Ifthe coin actually did come up heads, this is the end of theexperiment.

I If the coin came up tails, SB is injected with a special drugcocktail that will put her back to sleep and make hercompletely forget Monday’s events.

I She is then awoken on Tuesday morning (which, to her, seemslike Monday morning) and asked “From your point of view,what is the probability the coin came up heads?”

I If you were in SB’s shoes, how would you answer?

The Case of Sleeping Beauty

I A subject known as SB takes part in an experiment at theACME probability labs. She has full knowledge of all details ofthe experiment.

I Sunday night, SB goes to bed. While she is asleep, a coin isflipped.

I She is awoken Monday morning, and asked, “From your pointof view, what is the probability the coin came up heads?”. Ifthe coin actually did come up heads, this is the end of theexperiment.

I If the coin came up tails, SB is injected with a special drugcocktail that will put her back to sleep and make hercompletely forget Monday’s events.

I She is then awoken on Tuesday morning (which, to her, seemslike Monday morning) and asked “From your point of view,what is the probability the coin came up heads?”

I If you were in SB’s shoes, how would you answer?

The Case of Sleeping Beauty

I A subject known as SB takes part in an experiment at theACME probability labs. She has full knowledge of all details ofthe experiment.

I Sunday night, SB goes to bed. While she is asleep, a coin isflipped.

I She is awoken Monday morning, and asked, “From your pointof view, what is the probability the coin came up heads?”. Ifthe coin actually did come up heads, this is the end of theexperiment.

I If the coin came up tails, SB is injected with a special drugcocktail that will put her back to sleep and make hercompletely forget Monday’s events.

I She is then awoken on Tuesday morning (which, to her, seemslike Monday morning) and asked “From your point of view,what is the probability the coin came up heads?”

I If you were in SB’s shoes, how would you answer?

The Case of Sleeping Beauty

I A subject known as SB takes part in an experiment at theACME probability labs. She has full knowledge of all details ofthe experiment.

I Sunday night, SB goes to bed. While she is asleep, a coin isflipped.

I She is awoken Monday morning, and asked, “From your pointof view, what is the probability the coin came up heads?”. Ifthe coin actually did come up heads, this is the end of theexperiment.

I If the coin came up tails, SB is injected with a special drugcocktail that will put her back to sleep and make hercompletely forget Monday’s events.

I She is then awoken on Tuesday morning (which, to her, seemslike Monday morning) and asked “From your point of view,what is the probability the coin came up heads?”

I If you were in SB’s shoes, how would you answer?

The Case of Sleeping Beauty

I A subject known as SB takes part in an experiment at theACME probability labs. She has full knowledge of all details ofthe experiment.

I Sunday night, SB goes to bed. While she is asleep, a coin isflipped.

I She is awoken Monday morning, and asked, “From your pointof view, what is the probability the coin came up heads?”. Ifthe coin actually did come up heads, this is the end of theexperiment.

I If the coin came up tails, SB is injected with a special drugcocktail that will put her back to sleep and make hercompletely forget Monday’s events.

I She is then awoken on Tuesday morning (which, to her, seemslike Monday morning) and asked “From your point of view,what is the probability the coin came up heads?”

I If you were in SB’s shoes, how would you answer?

The Case of Sleeping Beauty

I A subject known as SB takes part in an experiment at theACME probability labs. She has full knowledge of all details ofthe experiment.

I Sunday night, SB goes to bed. While she is asleep, a coin isflipped.

I She is awoken Monday morning, and asked, “From your pointof view, what is the probability the coin came up heads?”. Ifthe coin actually did come up heads, this is the end of theexperiment.

I If the coin came up tails, SB is injected with a special drugcocktail that will put her back to sleep and make hercompletely forget Monday’s events.

I She is then awoken on Tuesday morning (which, to her, seemslike Monday morning) and asked “From your point of view,what is the probability the coin came up heads?”

I If you were in SB’s shoes, how would you answer?

The Pictoral Version

The Pictoral Version

The Pictoral Version

Monday:

The Pictoral Version

Monday:

Tuesday:

The Pictoral Version

Monday:

Tuesday:

Question: From SB's point of view, what is the probability the coin came up heads?

The Pictoral Version

Monday:

Tuesday:

Question: From SB's point of view, what is the probability the coin came up heads?

I Each of these three possible awakenings seem identical to SB.

I Ideas?

The Pictoral Version

Monday:

Tuesday:

Question: From SB's point of view, what is the probability the coin came up heads?

I Each of these three possible awakenings seem identical to SB.

I Ideas?

The Argument of David Lewis, the Halfer

Monday:

Tuesday:

I Let p be SB’s current estimation of the probability the coin isheads. Certainly, Sunday night p = 1/2.

I (?) In order for p to change, SB must gain new, relevantinformation.

I SB knows she will be awakened, hence by being woken up shegains no new information.

I Hence, during the experiment p = 1/2.

The Argument of David Lewis, the Halfer

Monday:

Tuesday:

I Let p be SB’s current estimation of the probability the coin isheads. Certainly, Sunday night p = 1/2.

I (?) In order for p to change, SB must gain new, relevantinformation.

I SB knows she will be awakened, hence by being woken up shegains no new information.

I Hence, during the experiment p = 1/2.

The Argument of David Lewis, the Halfer

Monday:

Tuesday:

I Let p be SB’s current estimation of the probability the coin isheads. Certainly, Sunday night p = 1/2.

I (?) In order for p to change, SB must gain new, relevantinformation.

I SB knows she will be awakened, hence by being woken up shegains no new information.

I Hence, during the experiment p = 1/2.

The Argument of David Lewis, the Halfer

Monday:

Tuesday:

I Let p be SB’s current estimation of the probability the coin isheads. Certainly, Sunday night p = 1/2.

I (?) In order for p to change, SB must gain new, relevantinformation.

I SB knows she will be awakened, hence by being woken up shegains no new information.

I Hence, during the experiment p = 1/2.

The Argument of David Lewis, the Halfer

Monday:

Tuesday:

I Let p be SB’s current estimation of the probability the coin isheads. Certainly, Sunday night p = 1/2.

I (?) In order for p to change, SB must gain new, relevantinformation.

I SB knows she will be awakened, hence by being woken up shegains no new information.

I Hence, during the experiment p = 1/2.

The Standard Argument of a Thirder

Monday:

Tuesday:

I Suppose the experiment is repeated 20 times.

I On average, we’d expect roughly 10 awakenings to be withthe coin heads, and 20 awakenings to be with the coin tails,for a total of 30 awakenings.

I SB has no reason to think a given awakening is more likelythan another, hence p = 10/30 = 1/3.

The Standard Argument of a Thirder

Monday:

Tuesday:

I Suppose the experiment is repeated 20 times.

I On average, we’d expect roughly 10 awakenings to be withthe coin heads, and 20 awakenings to be with the coin tails,for a total of 30 awakenings.

I SB has no reason to think a given awakening is more likelythan another, hence p = 10/30 = 1/3.

The Standard Argument of a Thirder

Monday:

Tuesday:

I Suppose the experiment is repeated 20 times.

I On average, we’d expect roughly 10 awakenings to be withthe coin heads, and 20 awakenings to be with the coin tails,for a total of 30 awakenings.

I SB has no reason to think a given awakening is more likelythan another, hence p = 10/30 = 1/3.

The Standard Argument of a Thirder

Monday:

Tuesday:

I Suppose the experiment is repeated 20 times.

I On average, we’d expect roughly 10 awakenings to be withthe coin heads, and 20 awakenings to be with the coin tails,for a total of 30 awakenings.

I SB has no reason to think a given awakening is more likelythan another, hence p = 10/30 = 1/3.

Another Halfer argument

Monday:

Tuesday:

I According to a thirder, p = 1/2 on Sunday, and p = 1/3 onMonday.

I However, SB not only gained no new information, there wasno funny business with forgetfulness drugs at that point. Howcould a rational person with no new information and with nocognitive lapses change her probability estimation?

Another Halfer argument

Monday:

Tuesday:

I According to a thirder, p = 1/2 on Sunday, and p = 1/3 onMonday.

I However, SB not only gained no new information, there wasno funny business with forgetfulness drugs at that point. Howcould a rational person with no new information and with nocognitive lapses change her probability estimation?

Another Halfer argument

Monday:

Tuesday:

I According to a thirder, p = 1/2 on Sunday, and p = 1/3 onMonday.

I However, SB not only gained no new information, there wasno funny business with forgetfulness drugs at that point. Howcould a rational person with no new information and with nocognitive lapses change her probability estimation?

The Argument of Adam Elga, a thirder

I Suppose instead the coin flip happened Monday night, afterSB had awakened once. (This should have no substantiveeffect on the experiment.)

Monday:

Tuesday:

I If SB is told it is Monday, then p = 1/2, since the coin flip isa future event.

I Let’s use Bayes’ rule to figure out the consequences of thisassertion.

The Argument of Adam Elga, a thirder

I Suppose instead the coin flip happened Monday night, afterSB had awakened once. (This should have no substantiveeffect on the experiment.)

Monday:

Tuesday:

I If SB is told it is Monday, then p = 1/2, since the coin flip isa future event.

I Let’s use Bayes’ rule to figure out the consequences of thisassertion.

The Argument of Adam Elga, a thirder

I Suppose instead the coin flip happened Monday night, afterSB had awakened once. (This should have no substantiveeffect on the experiment.)

Monday:

Tuesday:

I If SB is told it is Monday, then p = 1/2, since the coin flip isa future event.

I Let’s use Bayes’ rule to figure out the consequences of thisassertion.

The Argument of Adam Elga, a thirder

I Suppose instead the coin flip happened Monday night, afterSB had awakened once. (This should have no substantiveeffect on the experiment.)

Monday:

Tuesday:

I If SB is told it is Monday, then p = 1/2, since the coin flip isa future event.

I Let’s use Bayes’ rule to figure out the consequences of thisassertion.

The Argument of Adam Elga, a thirder

I Suppose instead the coin flip happened Monday night, afterSB had awakened once. (This should have no substantiveeffect on the experiment.)

Monday:

Tuesday:

I If SB is told it is Monday, then p = 1/2, since the coin flip isa future event.

I Let’s use Bayes’ rule to figure out the consequences of thisassertion.

The Argument of Adam Elga, a thirder

Monday:

Tuesday:

I Bayes’ Rule is

P(A | B) =P(B | A)P(A)

P(B).

I Let A = Coin is Heads and B = It is Monday.

I Bayes’ Rule says

P(Heads | Monday) =P(Monday | Heads)P(Heads)

P(Monday).

The Argument of Adam Elga, a thirder

Monday:

Tuesday:

I Bayes’ Rule is

P(A | B) =P(B | A)P(A)

P(B).

I Let A = Coin is Heads and B = It is Monday.

I Bayes’ Rule says

P(Heads | Monday) =P(Monday | Heads)P(Heads)

P(Monday).

The Argument of Adam Elga, a thirder

Monday:

Tuesday:

I Bayes’ Rule is

P(A | B) =P(B | A)P(A)

P(B).

I Let A = Coin is Heads and B = It is Monday.

I Bayes’ Rule says

P(Heads | Monday) =P(Monday | Heads)P(Heads)

P(Monday).

The Argument of Adam Elga, a thirder

Monday:

Tuesday:

I Bayes’ Rule is

P(A | B) =P(B | A)P(A)

P(B).

I Let A = Coin is Heads and B = It is Monday.

I Bayes’ Rule says

P(Heads | Monday) =P(Monday | Heads)P(Heads)

P(Monday).

The Argument of Adam Elga, a thirder

P(Heads | Monday) =P(Monday | Heads)P(Heads)

P(Monday).

I We already argued that P(Heads | Monday) = 1/2.

I Clearly P(Monday | Heads) = 1.

I I’m not going to claim exactly what P(Monday) is, butcertainly it is less than 1 (i.e. it could be Tuesday).

I P(Heads) = p is what we’re looking for.

I Hence1

2=

1 · psomething < 1

.

I Solving for p, we get p < 1/2.

The Argument of Adam Elga, a thirder

P(Heads | Monday) =P(Monday | Heads)P(Heads)

P(Monday).

I We already argued that P(Heads | Monday) = 1/2.

I Clearly P(Monday | Heads) = 1.

I I’m not going to claim exactly what P(Monday) is, butcertainly it is less than 1 (i.e. it could be Tuesday).

I P(Heads) = p is what we’re looking for.

I Hence1

2=

1 · psomething < 1

.

I Solving for p, we get p < 1/2.

The Argument of Adam Elga, a thirder

P(Heads | Monday) =P(Monday | Heads)P(Heads)

P(Monday).

I We already argued that P(Heads | Monday) = 1/2.

I Clearly P(Monday | Heads) = 1.

I I’m not going to claim exactly what P(Monday) is, butcertainly it is less than 1 (i.e. it could be Tuesday).

I P(Heads) = p is what we’re looking for.

I Hence1

2=

1 · psomething < 1

.

I Solving for p, we get p < 1/2.

The Argument of Adam Elga, a thirder

P(Heads | Monday) =P(Monday | Heads)P(Heads)

P(Monday).

I We already argued that P(Heads | Monday) = 1/2.

I Clearly P(Monday | Heads) = 1.

I I’m not going to claim exactly what P(Monday) is, butcertainly it is less than 1 (i.e. it could be Tuesday).

I P(Heads) = p is what we’re looking for.

I Hence1

2=

1 · psomething < 1

.

I Solving for p, we get p < 1/2.

The Argument of Adam Elga, a thirder

P(Heads | Monday) =P(Monday | Heads)P(Heads)

P(Monday).

I We already argued that P(Heads | Monday) = 1/2.

I Clearly P(Monday | Heads) = 1.

I I’m not going to claim exactly what P(Monday) is, butcertainly it is less than 1 (i.e. it could be Tuesday).

I P(Heads) = p is what we’re looking for.

I Hence1

2=

1 · psomething < 1

.

I Solving for p, we get p < 1/2.

The Argument of Adam Elga, a thirder

P(Heads | Monday) =P(Monday | Heads)P(Heads)

P(Monday).

I We already argued that P(Heads | Monday) = 1/2.

I Clearly P(Monday | Heads) = 1.

I I’m not going to claim exactly what P(Monday) is, butcertainly it is less than 1 (i.e. it could be Tuesday).

I P(Heads) = p is what we’re looking for.

I Hence1

2=

1 · psomething < 1

.

I Solving for p, we get p < 1/2.

The Argument of Adam Elga, a thirder

P(Heads | Monday) =P(Monday | Heads)P(Heads)

P(Monday).

I We already argued that P(Heads | Monday) = 1/2.

I Clearly P(Monday | Heads) = 1.

I I’m not going to claim exactly what P(Monday) is, butcertainly it is less than 1 (i.e. it could be Tuesday).

I P(Heads) = p is what we’re looking for.

I Hence1

2=

1 · psomething < 1

.

I Solving for p, we get p < 1/2.

Response by David Lewis

P(Heads | Monday) =P(Monday | Heads)P(Heads)

P(Monday).

I David Lewis, who insists that P(Heads) = 1/2, says the Elgaargument actually leads to the counterintuitive result thatP(Heads | Monday) = 2/3, even though the coin has not yetbeen flipped!

I Who do you side with?

Response by David Lewis

P(Heads | Monday) =P(Monday | Heads)P(Heads)

P(Monday).

I David Lewis, who insists that P(Heads) = 1/2, says the Elgaargument actually leads to the counterintuitive result thatP(Heads | Monday) = 2/3, even though the coin has not yetbeen flipped!

I Who do you side with?

Response by David Lewis

P(Heads | Monday) =P(Monday | Heads)P(Heads)

P(Monday).

I David Lewis, who insists that P(Heads) = 1/2, says the Elgaargument actually leads to the counterintuitive result thatP(Heads | Monday) = 2/3, even though the coin has not yetbeen flipped!

I Who do you side with?

Variations

The Case of Cloning SB

The Setup

Just like Classic SB, but instead of beingwoken up twice in the case of tails, SB iscloned, and each copy of SB is asked whatis the probability the coin came up heads.

AnalysisMost people agree in this case that p = 1/2is the right answer. Some halfers argue thatthis case is identical to Classic SB.

Variations

The Case of Many Awakenings

The Setup

Instead of being awoken just twice in thecase of tails, SB is awoken a hundred dif-ferent times, each time thinking it is stillMonday.

AnalysisA thirder is forced to believe p = 1/101during the experiment, which highlights thecounterintuitiveness of the thirder position.

Variations

The Case of Gambling SB # 1

The Setup

Every day SB is awakened, a gambler comesby and allows SB to wager, even money, thatthe coin came up tails. Should she take thebet?

Analysis

Obviously she should. By always taking thebet, she has a 50% chance of losing, and a50% chance of winning, but if she wins she’lleffectively be paid twice. So she comes outahead.

Variations

The Case of Gambling SB # 2

The Setup

Every day SB is awakened, a gambler comesby and allows SB to wager, even money, thatthe coin came up tails. However, she will bepaid later, and if she is offered the bet twice,one of the two decisions at random will betaken to be her decision.

Analysis

Now the bet is no longer in her favor. Thesetwo gambling cases give credence to theproposition that neither the halfers or third-ers are right, but it depends on how thequestion is asked.

References

I Elga, Adam. Self-locating belief and the Sleeping Beautyproblem. Analysis, 60: 143-147, 2000.

I Lewis, David. Sleeping Beauty: reply to Elga. Analysis, 61:171-176, (2001).

I Wedd, Nick. Some “Sleeping Beauty” postings.http://www.maproom.co.uk/sb.html