Inspired by "some" recent movie I saw, imagine the following scenario:
I sit with my friends Adam (A) and Ben (B) in a restaurant, when some accident happens that kills (0) everyone with the probability of 1/2. I am aware of this probability.
Luckily I survive (1), but hurt my eyes during accident so I cannot see who of my friends made it as well. Chances are 1/4 for each outcome A1B1, A1B0, A0B1, A0B0. [Knowledge state (KS) 1]
After a few minutes, I stumble over a person on the floor, which I can conclude to be definitely one of my friends and being alive, but I do not know who exactly. The person on the floor is either Adam or Ben. This rules out outcome A0B0.
If I now think about the chances that both of my friends survived this accident. I conclude it to be 1/3 as I have to be in one of the possible 3 remaining states of the world: A1B1, A1B0, A0B1 [KS 2]
Upon touching the person on the floor I am able to identify the friend as either Ben, who has a long beard, or Adam, who always shaves properly. The moment I am able to identify the person I know I am in either of two states:
1. The guy on the floor is Adam, so we have to be in state: A1B1 or A1B0 [KS 31]. The chances that we all made it is 1/2.
2. The guy on the floor is Ben, so we have to be in state: A1B1 or A0B1 [KS 32]. The chances that we all made it is 1/2.
No matter who it is on the floor, the moment I can identify him as either Adam or Ben, the chances that we all made it changes from 1/3 to 1/2, even though I know with 100% certainty in NS 2 that I have to be in either in KS 31 or 32.
I am not so sure why this somewhat hurts my head. In KS2 we simply ask: given that we know that at least one of two coins show heads, what is the probability of both showing heads. In KS 31/32 I ask after the first coin showed heads, whats the probability that the second coin shows heads as well.
It is however just weird to think about this scenario where I find a guy, knowing it has to be either Adam or Ben, and in the moment I know who it is exactly, I can be somewhat less concerned about our little group of friends.
At face value, that makes little sense to me and hence I expect something to be wrong, but I can't see what it is and I was unable to google this question.
If anyone could enlighten me I'd be most grateful.
Thanks a lot in advance.
Jan
I sit with my friends Adam (A) and Ben (B) in a restaurant, when some accident happens that kills (0) everyone with the probability of 1/2. I am aware of this probability.
Luckily I survive (1), but hurt my eyes during accident so I cannot see who of my friends made it as well. Chances are 1/4 for each outcome A1B1, A1B0, A0B1, A0B0. [Knowledge state (KS) 1]
After a few minutes, I stumble over a person on the floor, which I can conclude to be definitely one of my friends and being alive, but I do not know who exactly. The person on the floor is either Adam or Ben. This rules out outcome A0B0.
If I now think about the chances that both of my friends survived this accident. I conclude it to be 1/3 as I have to be in one of the possible 3 remaining states of the world: A1B1, A1B0, A0B1 [KS 2]
Upon touching the person on the floor I am able to identify the friend as either Ben, who has a long beard, or Adam, who always shaves properly. The moment I am able to identify the person I know I am in either of two states:
1. The guy on the floor is Adam, so we have to be in state: A1B1 or A1B0 [KS 31]. The chances that we all made it is 1/2.
2. The guy on the floor is Ben, so we have to be in state: A1B1 or A0B1 [KS 32]. The chances that we all made it is 1/2.
No matter who it is on the floor, the moment I can identify him as either Adam or Ben, the chances that we all made it changes from 1/3 to 1/2, even though I know with 100% certainty in NS 2 that I have to be in either in KS 31 or 32.
I am not so sure why this somewhat hurts my head. In KS2 we simply ask: given that we know that at least one of two coins show heads, what is the probability of both showing heads. In KS 31/32 I ask after the first coin showed heads, whats the probability that the second coin shows heads as well.
It is however just weird to think about this scenario where I find a guy, knowing it has to be either Adam or Ben, and in the moment I know who it is exactly, I can be somewhat less concerned about our little group of friends.
At face value, that makes little sense to me and hence I expect something to be wrong, but I can't see what it is and I was unable to google this question.
If anyone could enlighten me I'd be most grateful.
Thanks a lot in advance.
Jan
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