The “Sleeping Beauty Problem” is a question that has puzzled mathematicians for a long time, with multiple possible approaches and solutions. It initially appears to be a simple problem; however, it requires knowledge of decision theory to solve.
Decision theory is a subpart of probability theory which is used to find the probability of an event occurring by assigning individual probabilities to factors which influence the outcome. It was formulated in the mid-1980s by Arnold Zuboff and has been a debate ever since.
A Summary
A test subject (in this context, “Sleeping Beauty”) goes to sleep on a Sunday and is woken up once or twice depending on the toss of a fair coin.
If the toss’s outcome is “HEADS” she will be woken up on Monday. If the outcome is “TAILS”, she’ll be woken up on Monday, given a drug to forget that she was awake and woken up once more on Tuesday.
After waking her up on either of the days she will be asked “What is the probability that the flip resulted in a “HEADS” outcome?”
This problem can be solved using the:
1. Thirder Position
2. Halfer Position
Thirder position:
[P(H) = 1/3 ] & [P(T) = 2/3 ]
This approach was devised by Adam Selga, who said that the probability of “HEADS” is .
He proved his conclusion by stating that the test subject would believe that:
[P(T, Mon) = P(T, Tue)]
i.e., the probability of the day being a Monday or Tuesday is equal if the outcome is “TAILS”.
If the test subject wakes up believing that it is a Monday:
[P(H,Mon) = P(T,Mon)]
i.e., the probability of “HEADS” and “TAILS” will be equal (since the coin is fair).
By equating the 2 statements:
[P(H,Mon) = P(T,Mon) = P(T,Tue)]
It can be concluded that all 3 outcomes are equally favorable, therefore the probability of a “TAILS” is twice that of a “HEADS”.
A simpler way to understand this is by solving the Monty Hall problem.
The Monty Hall Problem: A Summary
Assume there are 3 doors: one containing $100,000, while the other 2 contain $5. After picking a door, one of the remaining doors (containing $5) is removed, and you are provided an option to switch to another door. What is the best choice in order to get the $100,000?
In this case, switching is the best choice since the probability of the chosen door remains as 1/3, however the leftover door’s probability of containing the $100,000 rises to 2/3.
Halfer Position:
Proposed by David Lewis as a response to the Thirder Position, the Halfer Position claims that the probability of “HEADS” and “TAILS” are equal:
[P(H) = P(T) = 1/2 ]
Since the test subject has no information provided on whether it is a Monday or a Tuesday, the Halfer Position suggests that assuming [P(H,Mon)] = 1/3] is invalid and contradicts the idea of the question. From the test subject’s perspective, the day is an unknown factor, therefore the number of times that she’s woken up is also unknown.
Conclusion:
Due to the framing of the question, it is rather difficult to determine one single answer for whether the answer is 1/3 or 1/2 , leading to the question’s paradoxical and puzzling nature.
By: Mohammad Umair Bamboowala, AS A
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