The paradox of sleeping beauty is a dilemma that questioned our certainties on Calculation of probability. Let’s imagine this situation: the beauty falls asleep on Sunday and is woken up on Monday or Tuesday based on the launch of a coin – if it comes out it will be woken up on Monday, if Tuesday comes out on Tuesday. Upon awakening, she is asked how much it believes it is likely to come out in the launch of the coin. It seems immediate to think that the probability is 50-50, because the possibilities are two: head or cross. Yet, if instead of focusing on the coin we focus on possible cases of awakening, we realize that they become three possible cases, and therefore the probability is no more than 50%!
Which of the two readings is the right one? We don’t know: i mathematicians have not yet found an agreement on this paradox. Let’s see the details of the dilemma and what are the main currents of thought on its solution.
The paradox of the sleeping beauty and the launch of the coin
Sleeping beauty is involved in an experiment in which it is falling asleep on Sunday with a sleeping pill for being awakened on Monday or Tuesday depending on the outcome of a single coin launch. The rules of the experiment are these:
- If he comes out The girl is woken up on Monday and a question is asked;
- If Croce comes out The girl is woken up on Monday, she is asked for the same question, then she falls asleep again with the sleeping pill to wake her up then on Tuesday and ask her the same question again.
We keep in mind that the sleeping pill causes a partial loss of memory, so with each awakening the sleeping beauty does not remember if it has already been awakened previously, but recalls the rules of the experiment. The question that is asked each awakening is:
How much you think it is likely that in the launch of the coin it came out head?
It looks like a obvious question: the coin every time can only have two outcomes, head or cross, and therefore the probability will be 50%. Yet, a probabilistic dilemma is born: if we considered the “currency” variable, but the “awakening” variable, the possible cases become three! And therefore the probability changes. Let’s see why.
The two possible solutions to the paradox of sleeping beauty: what is the probability?
The most immediate answer to which many come is that the sleeping beauty will reply that the probability that it has come out is 50%, because the possible results of the coin are only two – head or cross – and therefore each of the two has 50% probability of appearing, and for this reason the first solution is that the sleeping beauty replies:
The probability that during the experiment has come out is 50% .
But if we change the point of view, things get complicated a lot, but the answer changes above all! Instead of focusing on the coin, in fact, we now focus on the Awakens. With this change of perspective, the number of possible results changes! The situations in which the sleeping beauty could find at the time of awakening are no longer simply two, but are three, Because in the case of the exit of the cross there are two possible scenarios, both that of Monday and Tuesday:
- It may think you have been woken up about Monday because it came out head
- or she was woken up of Monday but it came out cross And it will be re -deduced
- Finally, it may be the awakening Tuesday because it initially came out cross
This time the possible cases are 3, but only one of the three provides that it has come out, so according to this perspective, the probability that the head has come out is one in three cases, that is 1/3 which is written in percentage is about 33.33%. According to this reasoning, therefore, the response of the sleeping beauty will be:
There probability that it has come out is 1/3, i.e. about 33%
Both reasoning are correctyet they give different results! So, what is the right answer? The argument that leads to conclude that the right answer is 1/3 is strengthened by the philosopher of science Adam Elga which expressed it formally using the calculation of the conditioned probability, a complicated way of calculating the probability when an event depends on another event. But if we think about it, the coin could be launched even before even the start of the experiment and would not change anything, so the chances of the head would still be 50% regardless of the awakenings of our protagonist.
In short, it is really a nice puzzle, and in fact There does not seem to be a definitive agreement on the subjectas evidenced by the fact that there are dozens of publications to support one or the other theory.
