In the "Sleeping beauty problem", would Elga's answer change if it's 7 days instead of 2?

Question

Consider this Sleeping Beauty-like scenario: Some researchers are going to put Sleeping Beauty to sleep. During the seven days that the experiment will last, they will trie to wake her up either once or seven times, depending on the toss of a fair coin (Heads: once; Tails: seven times). After each waking, they will put her to back to sleep with a drug that makes her forget that waking. When she is first awakened, to what degree ought she to believe that the outcome of the coin toss is Heads?

If we follow Elga's reasoning what is the answer here? Give some justication.

Considering Elga answers in the case of tails having her woken up 2 times, he says probabilites are 1/3, I think he'd answer 1/8 in this case as P(H on day 1)=P(T on day 1) and P(T on day 1)=P(T on day 2)=...=P(T on day 7).

Therefore, overall there are 8 options and with no room for discrimination he'd assign 1/8 to each. Is this reasoning correct?

Answer 1

As you probably know, there is no consensus which is the right solution to the original problem. Some people, including Adam Elga say it is 1/3, other people including David Lewis say it is 1/2.

If you're asking what the result of Elga's method would be, you're right, it would be 1/8. But I would write "P(H and day 1)" not "P(H on day 1)", that can be confused with the conditional probability "P(H | day 1)".

Answer 2

The question itself revolves around a question of knowledge and belief; what does Princess Aurora know and how much weight she should place on what she has been told?

Let us say,

1. If she has been told nothing about the experiment other than the coin is fair ie how often she would be awoken, she will simply say half - which simply reflects her knowledge of the fairness of the coin. This will be different from the experimenters, who knowing more, can say more, and they will say 1/8th; and they can prove it by pointing to the recordings that they have made of many runs and reruns of this experiment.

2. But if the princess knows knows a little more - that the drug wipes out any memory of any past awakenings - then she can say a little more: that she can't prove by her own experience of wakenings and reawakenings that she ought to say a half.

3. But all this is of course dependent on what degree of trust Princess Aurora has in the experimenters; let us say, for what ever reason, a degree of suspicion has been awoken in the princess - she might have overheard something that didn't sound quite right - and being a resourceful princess she makes a small mark with a fingernail on the chest beside her bed every time she is awoken; she will, then straightaway find out that she is being awoken more often than she can remember; but from this she cannot determine probabilities between the two cases; what she can confirm is that there is more to the experiment than she has been told.

Answer 3

Let's do this experiment on a large scale. Let's take 200 "sleeping beauties"; 100 are woken up seven times, and 100 are woken up only once. It happens 800 times that a sleeping beauty wakes up. 700 times it is one of the 100 that are woken up 7 times. 100 times it is one of the 100 that are woken up only once.

So if you wake up the chance is 7/8th that you are one of the 100 that are woken up 7 times. Well, that's the situation if Sleeping Beauty is given drugs so that she forgets she has woken up before.

Without the drugs, 100 times she is one of the 100 woken up only once, 100 times she is woken up for the first time out of seven times, and 600 times she knows that she woke up before. In that case, being woken up for the first time the chance are 50/50.

There is a nice strawman argument saying that when she is woken up, the probability is 8/14th that it is the first time, and 1/14th each that it is the second, third, ..., seventh time. Perfectly correct but it isn't the question that was asked. What was asked was what she should believe about the initial dice throw.

If she is allowed to make a bet for $1,000 each time she wakes up, what should she bet? Since she doesn't know how often she has been woken up, she should make the same decision each time. If she says heads, then the initial chance that heads was thrown is 50%. However, with tails she gets woken up 7 times. So saying "heads" gives her a 50% chance of winning once, but also a 50% chance of losing seven times. Saying tails gives her a 50% chance of losing once, but also a 50% chance of winning seven times.

She is woken up 7 times more often when the result is tails. So when she is woken up, the chance is much higher that the result is tails.

PS. If she is woken up the first time, and she is reliably told that she was woken up the first time, then heads and tails have an equal 50% chance.