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:

$\frac{1}{3}$

to find:

the probability of the coin.

solution:

lie:

$( \frac{1}{3} ) \times ( \frac{1}{3}) \times ( \frac{1}{3} )$

$= \frac{1}{27}$

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

truth:

$( \frac{2}{3} ) \times ( \frac{2}{3} ) \times ( \frac{2}{3} )$

$= \frac{8}{27}$

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

probability:

$( \frac{8}{27} ) \div (( \frac{8}{27} ) + ( \frac{1}{27} ))$

$= \frac{8}{9}$

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