Sleeping Beauty Problem
If you are interested in challenging your brain, this probability problem is for you. To see how paradoxes occur, keep reading!
History
Sleeping Beauty paradox is a thought experiment in decision theory. The problem was first formulated in an unpublished work in the mid-1980s by Arnold Zuboff, who is an American philosopher. The name “Sleeping Beauty” was given to the problem by Robert Stalnaker. It shows similarity with Bertrand Paradox as both of them have several answers according to different approaches.
The Problem
From Wikipedia:
Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details: On Sunday she will be put to sleep. Once or twice, during the experiment, Sleeping Beauty will be awakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes her forget that awakening. A fair coin will be tossed to determine which experimental procedure to undertake:
If the coin comes up heads, Sleeping Beauty will be awakened and interviewed on Monday only.
If the coin comes up tails, she will be awakened and interviewed on Monday and Tuesday.
In either case, she will be awakened on Wednesday without interview and the experiment ends.
Any time Sleeping Beauty is awakened and interviewed she will not be able to tell which day it is or whether she has been awakened before. During the interview Sleeping Beauty is asked: “What is your credence now for the proposition that the coin landed heads?”
There are two approaches to the problem. Some mathematicians believe the correct answer is 1/2 and some other mathematicians believe the answer is 1/3.
Halfer Position
Before the experiment, the probability of the coins landing heads is 1/2. After waking up, Sleeping Beauty still does not know what the day it is or whether she has been awakened before, thus there is not any more information she gains or any information she loses. “While sleeping beauty being woken multiple time may allow others to gain information from the waking, her memory loss does not allow her to do so.” As a result, nothing has changed according to Sleeping Beauty so there is no need to change the probability which is 1/2.
Thirder Position
During the interview three cases are possible. 1) It is Monday and the coin has landed heads. 2) It is Monday and the coin has landed tails. 3) It is Tuesday and the coin has landed tails. When the coin landed tails, both cases 2 and 3 will occur, so we can say P(2) = P(3). On the other hand, on Monday two cases are possible: 1 and 2. The probability of the first case and the second case are the same as the coin is genuine. Now, we have:
P(1) = P(2), P(2) = P(3) ⇒ P(1) = P(2) = P(3) and P(1) + P(2) + P(3) = 1 so, P(1) = P(2) = P(3) = 1/3.
In other words, in two cases, case 2 and case 3, the coin has landed tails and in case 1 the coin has landed heads. As a result, we expect the probability to be 1/3.
For further judgements, you can check this link out.
Let’s change the question into a more accurate problem.
Professor tosses the coin and asks Sleeping Beauty “Heads or tails?” Whenever Sleeping Beauty gives the right answer the game ends. If the answer is wrong two cases are possible:
1) The coin landed heads and SB said tails.
2) The coin landed tails and SB said heads.
In the first case, the professor tosses the coin again and asks again. In the second case, the professor gives an amnesia-inducing drug to SB which makes her forget the previous answer she gave. After taking the drug professor asks again “Heads or tails?” if SB gives the right answer, which is tails, the game ends as mentioned. If SB gives the wrong answer then, professor tosses the coin again and implements the same process. What is the probability of SB’s giving the correct answer? In other words, what is the probability of the game’s being finite?
It is actually 1, the game will definitely end. In order to prove, let’s find its possibility of not ending, which should be 0. It is time to stop reading and to try to solve it. If you have tried something to figure the problem out, let’s go on with the solution.
STEP 1: Figure out the diagram.
STEP 2: Define the probability.
In order to find the probability of the game’s ending, we will try to find the probability of the game's not ending. It will work because there are only two possibilities: Game can have an end or game can be endless. So, P(infinite) + P(finite) = 1. Let p show the probability of the game’s being endless, which means P(infinite) = p.
STEP 3: Write p as the sum of other probabilities.
Before starting to calculations, let’s make it clear that P(heads) = P(tails) = 1/2.
As we are looking for the infinite case, we will consider the cases that SB gives the wrong answers. For example, if the coin has landed heads, SB should say tails in order not to finish the game. And the probability of that is,
P(heads).P(SB says tails.) = 1/2 x 1/2 = 1/4.
Now we can write p as the sum of other possibilities.
In every coin-tossing process, the same game starts again. Because of that, we can adjust our diagram a little bit. Now, we can say “When the game starts again?” means “When does the professor toss the coin again?”. By definition of p, whenever the game starts again the same probability occurs. This perspective gives us the equality which is,
p = P(Heads).P(“SB says tails.”).p + P(Tailes).P(“SB says heads.”).P(“SB says tails for the second time.”).p
If we write the possibilities, we have
p = 1/2.1/2.p + 1/2.1/2.1/2.p = 3/8.p ⇒ p=0.
We found that P(infinite) = p = 0. So P(finite) = 1.
In the same way, we can directly find P(finite). Let’s assume P(finite) = q. The plan is to write q as the sum of other possibilities. In this case, we want SB to say heads if it is heads and to say tales if it is tails. In the same way, every time the professor tosses the coin, the game starts again and the same probability occurs.
As written in the image, q = 5/8 + 3/8.q ⇒ q =1. As a result, the game has an end. Here is the more important question, “What is the probability of the game’s ending in a position in which the coin tossed heads?” You can find the probability using the same way which is going to be 2/5.
Resources,
1 The Sleeping Beauty Problem. Youtube. Web. 15.03.2020.
2 Sleeping Beauty Problem. Wikipedia. Web. 16.03.2020.
3 Some Sleeping Beauty Postings. maproom. Web. 16.03.2020. Link: http://www.maproom.co.uk/sb.html#arg5
