Anton speaks truth two out of five times. He throws a die and reports that it is a four. Find the probability that it is actually four.
a. 3/16
b. 2/7
c. 2/17
d. 7/30
Answers
Answer:
a . 3/16
Step-by-step explanation:
it is the correct answer
Step-by-step explanation:
There are (at time of writing), 8 other answers to this question, none of which is fully correct, complete and properly explained.
First, it is not correct to say that the probability is 2/5 simply because that is the probability that Anton is telling the truth. The conditional probability that the die shows 4 given that Anton says it does (which is what we want) is not in general the same as the converse, being the conditional probability that he tells the truth given that the die shows 4.
To see why, here is a similar problem: Suppose there are 100 gladiators, of whom exactly one is Spartacus. Each gladiator (whether Spartacus or not) has a 50% probability of saying he is Spartacus and a 50% probability of saying he is not Spartacus. Suppose a randomly chosen gladiator tells you he is Spartacus – what is the probability that he is Spartacus?
Answer: 1/100 (not 50%). This is easy to see as everyone is equally likely to be chosen and is equally likely to say he is Spartacus. The fact that Spartacus himself has a 50% probability of telling the truth does not mean that someone (randomly chosen from the 100 gladiators) who says he is Spartacus has a 50% probability of indeed being Spartacus.
Now back to the question:
“Anton speaks the truth two out of five times. He throws a die and reports that it is a four. What is the probability that it is actually a four?”
The question as stated is incomplete and cannot be solved. This is because it does not specify what Anton says if the die does not show a four and he does not tell the truth. For example, if it shows 5 and he does not tell the truth, what does he say? Does he always say “four”? Or might he say three or six, say?
Here are two possibilities.
(A)
If we assume that if the die shows anything but a four and Anton lies, then he always reports that it is a “four”, then the answers of Kavita Chawdhary and Riju Bhatt, who both obtain 217217, are correct. However, this assumption needs to be stated.
(B)
Alternatively, suppose we assume that, given that Anton does not tell the truth, then he picks one of the other 5 numbers to report with equal conditional probability of 1515each. So for example, if it shows a five and he lies, then he is equally likely to report that the die shows 1, 2, 3, 4, or 6.
In that case, I would refer you back to Kavita’s answer for the explanation using Bayes’ Theorem, but her formula then becomes
25⋅1625⋅16+35⋅15⋅5625⋅1625⋅16+35⋅15⋅56
This equals 2525 – but not for the reasons given by the other answers!
In either case, we had to make an assumption about what Anton reports if the die does not show a four. As this was not given in the question, it is not answerable.