Each of Abner, Berenice, Calvin and Dolores is either a knight who always tells the truth, or a liar who always speaks falsely, or a knave who alternates between lying and telling the truth in any order.
They say:
- Abner: Berenice is a liar.
- Berenice: I am a knight.
- Calvin: If asked, Berenice would say that she is a knight.
- Dolores: I am a knight.
Given that each of the four individuals being any of the 3 types is equally likely, and all the four statements are simultaneous and independent, determine the probability that:
- Abner is a knight, Berenice is a liar and each of Calvin and Dolores is a knight.
*** For an extra challenge, solve this puzzle without taking resort to a computer program/excel solver aided methodology.
My assumption concerning Calvin's statement is different than Charlie's. Since Calvin speaks independently of and simultaneously with Berenice, his statement cannot depend on what she has just said. Therefore, Berenice cannot be a knave. She is a knight or a liar, each with a 1/2 probability.
If Berenice is a liar, then Calvin is either a knight or a knave who is telling the truth. Since a knave only tells the triuth half the time, the conditional probability that Calvin is a knight is 2/3.
Similarly, if Berenice is a liar, then Abner is either a knight or a knave who is telling the truth. The conditional probability that Abner is a knight is 2/3.
Finally, Dolores is either a liar or a knight or a knave who is lying. Since a knave only lies half the time, the probability that Dolores is a knight is 2/5.
Altogether, the probability that Bernice is a liar and the other three are knights is (1/2)*(2/3)*(2/3)*(2/5) = 4/30, or a little more than 13%.
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OOPS! That is not how to multipy fractions. As Math Man points out,
(1/2)*(2/3)*(2/3)*(2/5)=8/90=4/45=0.0888888888... , and not 4/30
Edited on September 12, 2023, 8:09 pm
Extra Challenge Accepted (spoiler)
