Question

Three people, Adam, Becky and Charley, each lie 1/3 of the time at random, otherwise they tell the truth. A fair coin is flipped, and they all see it. They all say it is heads. What is the probability the coin was actually tails?

Answer 1

given:

`(1)/(3)`

to find:

the probability of the coin.

solution:

lie:

`( (1)/(3) ) * ( (1)/(3)) * ( (1)/(3) )`

`= (1)/(27)`

there's 1/27 chance of them to lie.

truth:

`( (2)/(3) ) * ( (2)/(3) ) * ( (2)/(3) )`

`= (8)/(27)`

there's 8/27 chance that all of them told the truth.

probability:

`( (8)/(27) ) / (( (8)/(27) ) + ( (1)/(27) ))`

`= (8)/(9)`

therefore, between those two chances, it's an 8/9 chance it's actually Heads.