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A Quick Probability Problem

Here I am with a short post. This is a short problem on probability. Post your solution in the comments section.

Calvin lies \(\frac{2}{3}\) of the time and Peter lies \(\frac{1}{3}\) of the time. I asked Calvin if he'd completed his homework. He said no. Then Peter said, "Calvin is telling the truth."

What is the probability that Calvin completed his homework? What is the probability that he didn't?

Oh yes, and "All characters appearing in this problem are fictitious. Any resemblance to real persons, living or dead, is purely coincidental." :)

Note by Mursalin Habib
3 years, 4 months ago

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By Peter's statement, the truth of Calvin's statement is the same as the truth of Peter's statement (ie they're both lying or both telling the truth). The probability that they are both lying is \(\frac{2}{3} \cdot \frac{1}{3} = \frac{2}{9}\). The probability that they are both telling the truth is \(\frac{1}{3} \cdot \frac{2}{3} = \frac{2}{9}\).

We know that one of these cases must be true, and thus the probability that Calvin didn't complete his homework is,

\[ \frac{^2/_9}{^2/_9 + {}^2/_9} = \boxed{\frac{1}{2}} \]

Is this right, and would this also work if the probabilities were unequal? Ben Frankel · 3 years, 4 months ago

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@Ben Frankel

\[\frac{2/9}{2/9}=\frac{1}{2}\]

What?!? Mursalin Habib · 3 years, 4 months ago

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@Mursalin Habib I wasn't thinking. It should make more sense now...

It was supposed to be the probability of the desired event, divided by the sum of the probabilities of all allowed possibilities. Ben Frankel · 3 years, 4 months ago

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@Ben Frankel I see what you did there. Can you find what's wrong with this argument?

Both Calvin & Peter said that Calvin didn't complete his homework. So, for the homework to be incomplete, both of their statements have to be true. But the probability that both of their statements are true is \(\frac{2}{9}\). So, our desired probability is \(\frac{2}{9}\). Mursalin Habib · 3 years, 4 months ago

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@Mursalin Habib That argument is wrong, because it misses information that can be derived from Calvin and Peter's statements, which would change the probability (from the reader's perspective).

Namely, it can be seen that one person lying while the other is telling the truth would be a logical fallacy, and so they are either both lying or both telling the truth. Thus, you must compare the probability of the desired event to the probabilities of all possible events, discarding the impossible ones. Ben Frankel · 3 years, 4 months ago

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@Ben Frankel Okay, now what's wrong with this argument?

If Calvin's statement is true, so is Peter's. The probability that Calvin's statement is true is \(\frac{1}{3}\). So, the probability that he didn't complete his homework is simply \(\frac{1}{3}\). Mursalin Habib · 3 years, 4 months ago

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Both have said the same answer "No", so either both are true or both are false. So, the event which has already occurred is that they have said the same thing which has the probability \(\frac{4}{9} (\frac{1}{3} * \frac{2}{3} + \frac{2}{3} * \frac{1}{3})\). What we are required is that both spoke the truth which has probability \(\frac{2}{3}\). So according to Bayes' theorem the answer is \(\frac{\frac{2}{9}}{ \frac{1}{9} + \frac{2}{9}} = \boxed{\frac{1}{2}}\) Ayush Pateria · 3 years, 4 months ago

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@Ayush Pateria Typo: It's 2/9 instead of 1/9 in the denomination. Ayush Pateria · 3 years, 4 months ago

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@Ayush Pateria denominator* Ayush Pateria · 3 years, 4 months ago

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