Doorle Offerhaus
♦️ ereliuer eteer ♦️Ava and Eve are identical twins, but Ava tells the truth with a probability of 0.75, and Eve tells the truth with a probability of 0.25 (independently of each other). Suppose, I meet them in the street and ask one of them: “Are you Ava or Eve?” and she responds “I'm Ava.” Then I ask the other girl: “Did she tell the truth?” and she responds “Yes.” What are the chances that both girls told the truth this time?
Ian Ramjohn@ereliuer_eteer If Ava told the truth, then Eve is telling the truth. So only the p. that Ava is telling the truth matters. So 0.75
@iramjohn But you don't know who of them is the real Ava.
notsoloud@ereliuer_eteer
You meet very interesting people. Unsettling, but very interesting.
You meet very interesting people. Unsettling, but very interesting.
cube!@ereliuer_eteer even though the odds of receiving a single true answer from both (with independent Qs) in one go is 3/16, we already know they were either both telling the truth or both lying (which happens to also have a 3/16 chance). our actual question is the *chance that we picked ava first,* which is a simple 1/2 chance.
Roger Crew✅❌☑🗸❎✖✓✔Is Eve, when lying about what her name is, constrained to only be able to answer "I'm Ava"? Or is there a non-zero probability she'll give some name entirely different from "Ava" or "Eve" (which seems to be allowed)?
(it's clear that this makes a difference since if that latter probability is 1, e.g.., she *always* says "I'm Susan" when she's lying about her name, then "I'm Ana" + "yes" is both girls telling the truth with probability 1)
@wrog She is not constrained, but you do know beforehand that she is either Ava or Eve. If she told the truth, it means she is Ava. If she lied, it means she is Eve.
Right, but we still need to know the probability that Eve will *say* she is Ava when she's lying. As things currently stand, all we have for the probabilities on Eve's possible responses to the first question are:
"I am Eve"
⟶ 0.25
(telling the truth)
"I am Ava"
⟶ 0.75x
(lying in one particular way)
"I am [neither Ava nor Eve]"
⟶ 0.75(1-x)
(lying some other way)
1/3
The two possible scenerios that result in "I am Ava" + "yes", answers are that
(1) Ava is giving the first answer (0.5*0.75*0.25) and both are telling the truth,
(2) Eve is giving the first answer (0.5*0.75x*0.25) and both are lying
which makes the overall conditional probability (that they're both being truthful given that the answers are "I am Ava" and "yes") is 1/(1+x)
so we need to know what x is.
seems to me that, for the problem to actually work (have a determinate answer), you want the first question to be, "Are you Ava?" rather than "What's your name?"
3/3
@wrog After thinking about it more (and doing some frequentist experiments), I agree that the assumption that the only possible answers to the first question are "Ava" and "Eve" is necessary. I didn't realize it when I wrote the problem. Thanks for pointing that out.
Alexander The 1st@ereliuer_eteer As I understand, it's 0.1875 (Or 0.19 essentially rounded up?), since while there's a decent chance that if Eve lies, Ava calls them out, there's also a large chance that if Ava tells the truth, Eve calls them out, despite Ava telling the truth.
Kim Reece50% assuming your choice of who to ask first was random.
If you first asked Eva, her answer was false (0.75), and Ana's reply was false (0.25).
If you first asked Ana, her answer was true (0.75) and Eva's reply was true (0.25).
Either way an equally likely (or unlikely) sequence of events occurred, with which chain of events determined by your initial choice.
Patrick Vanhoucke@ereliuer_eteer Since there is no dependency or relationship between what they say, I would say: the product of both chances, so 0.1875.
Jasper Klewer@ereliuer_eteer 50% of asking Ava * 75% of Ava truth * 25% of Eve truth = 9,375%. In all other cases you’re talking to Eve, or someone is lying (and provides another answer).
Sibrosan0.5
If the first girl is Ava, they both must have told the truth; if she's Eva, they must have both lied.
And the probability of each option is the same, 0.75 * 0.25 = 0.25 * 0.75
Menno@sibrosan @ereliuer_eteer
Ha! Your answer made me think it through. I think you're correct that it's 0.5. but it has nothing to do with their chance of telling the truth.
You started off right. The question is: what is the chance that both girls spoke the truth.
*If* the first girl is Ava, then they're both right. If the first girl is Eve, then she lied and it doesn't even matter what the 2nd girl said.
The chance of the first girl being Ava is 50/50.
Ha! Your answer made me think it through. I think you're correct that it's 0.5. but it has nothing to do with their chance of telling the truth.
You started off right. The question is: what is the chance that both girls spoke the truth.
*If* the first girl is Ava, then they're both right. If the first girl is Eve, then she lied and it doesn't even matter what the 2nd girl said.
The chance of the first girl being Ava is 50/50.
"but it has nothing to do with their chance of telling the truth."
I think you're mistaken.
Suppose, as an extreme example, that the chance for Eve to speak the truth is 1, instead of 0.25.
In that case, given that both confirm the first girl is Ava, the chance of both telling the truth is 1, not 0.5.
Jos DingjanPick a 1st girl to talk to at random: P_A = p; P_E = 1 - P_A = 1-p [1]
Case A: 1st girl speaks truth (she is A); 2nd girl speaks truth (1st girl spoke truth) > 2x true
Case E: 1st girl lies (she isn’t A); 2nd girl lies (1st girl didn’t speak truth) > 2x false.
No other scenarios compatible with the givens, so P_TT + P_FF = 1, P_TT = P_A = p and P_FF = P_E = 1 - p
If P_A = P_E then p = 0.5
[1] We could assume p = 0.5 given, but…
Jonathan👣🚲@ereliuer_eteer 1/2. Gotta love conditional probability. Edit: this is assuming Eve would always choose Ava as her fake name (certainly the most effective lie in our context). If not, the probability is greater.
@jonathanavt Yes, I agree without that assumption, the result can be different.
Peter@ereliuer_eteer Hi, I stumbled upon this question via @ionica. If I remember correctly my statistics course 40 years ago I think the answer is 0,1875. This is the result of 0,75*0,25.