Joe is a chronic liar and gives a true answer only 2 times out of 7. He throws a die and reports that it is a 5. What is the probability that it is actually a 5.

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Joe gives a true answer only 2 times when asked 7 questions. When a die is thrown he reports that it is a 5.

The probability of the number on a die turning up 5 is `1/6` and the probability of the number not being 5 is `5/6` . The probability of the number being 5 and Joe reporting it as 5 is `(1/6)*(2/7)` . The probability of the number not being 5 but Joe reporting it as 5 is `(5/6)*(5/7)`

If Joe reports that the number on a die is 5, the probability that it is actually 5 is `((1/6)*(2/7))/((1/6)*(2/7)+(5/6)*(5/7))`

=> `(2/42)/(2/42 + 25/42)`

=> `2/27`

**The probability that a die actually shows the number 5 if Joe says that it does is `2/27` .**

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