0

A man is known to speak the truth 3 out of 5 times. He throws a die, and reports that it is a 1. Find the probability that it is actually 1. (CBSE 2014)

Solution 1

Assume the die is fair. If the man throws the die 30 times, the expected outcome is each of {1,2,3,4,5,6} occurs 5 times. He $$\text{tells the truth } \frac{3}{5} \times30=18 \text{ times}, \text{lies} \frac{2}{5}\times 30=12 \text{times}$$

Tabulating, he gets a 1 five times. Out of those, he tells the truth 3 times. Out of the 25 times he does not get a 1, he lies 10 times. Those lies are equally distributed among the possible lies, and hence he lies that it is a "1" two times.

Apply the logic of Bayes Theorem. There are five outcomes with a 1, out of which 3 are represent what we are looking for. Hence, the final probability is 3/5.

Solution 2

Says it is $\frac{3}{13}$

The solutions are different, and the difference of course comes from the different assumptions being made here. Which assumption/solution is correct, and why? Or is it that the question needs to be made more precise?

Starlight
  • 1,680
  • 4
    These problems seldom provide enough information. I suppose we are meant to assume that when the person chooses to lie they do so uniformly amongst the $5$ possible false answers. – lulu Sep 16 '22 at 02:07
  • @lulu If that were the case, then Solution 1 is correct. But the solution found on multiple sites (I have linked one such in Solution 2), does not make that assumption. – Starlight Sep 16 '22 at 02:09
  • 1
    The "official" solution you link to states the nonsense equation $P(A,|,E_2)=\frac 25$ which I can't guess at an interpretation of. Are we to imagine that the man always says $1$ if it isn't $1$ and he chooses to lie? That's a truly bizarre assumption to make – lulu Sep 16 '22 at 02:12
  • It is not an "official" solution. It is the website's interpretation. But that same solution can be found in many places. – Starlight Sep 16 '22 at 02:13
  • Please provide some alternate links. – lulu Sep 16 '22 at 02:14
  • https://www.topperlearning.com/answer/a-man-is-known-to-speak-truth-3-out-of-5-times-he-throws-a-die-and-reports-that-it-is-a-one-find-the-probability-that-it-is-actually-a-one/vyn3bvll – Starlight Sep 16 '22 at 02:15
  • https://www.toppr.com/ask/question/a-is-known-to-speak-truth-3-times-out-of-5-times-he-throws-a-5/ – Starlight Sep 16 '22 at 02:15
  • sorry, all those sites are weird and have a lot of typos on them. I would not use them for references. – lulu Sep 16 '22 at 02:16
  • I have a book with past year questions from the exam. It has the same solution. Hence, this was likely the "official" solution (though I can't provide a reference for it now. That is precisely the reason why I brought up this question. My logic doesn't agree with the prevailing one. – Starlight Sep 16 '22 at 02:20
  • regardless: if the problem doesn't specify the rule the man follows when he chooses to lie the problem can not be answered. Different assumptions lead to different conclusions. – lulu Sep 16 '22 at 02:27
  • 2
    Might be worth adding: the answer is different if the problem states "the die is thrown and the man is asked if it is a $1$ to which he replies Yes". In that case I agree with the official solution. For binary questions, it is clear what "Lie" means. – lulu Sep 16 '22 at 02:46
  • I think that is the interpretation that the question wants. Due to historical reasons, the interpretation is clear to those in the system for many years, but not otherwise. Thanks. This was very useful. – Starlight Sep 16 '22 at 03:02
  • Would you agree that Solution 1 is correct, given the usual interpretation? – Starlight Sep 16 '22 at 03:02
  • I'm closing the question as a duplicate. The discussion on the linked duplicate is good and here is another duplicate with further discussion. As you'll see, the question generates lots of confusion but, to me at least, it's just because it is so vaguely worded. – lulu Sep 16 '22 at 11:36

0 Answers0