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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?

Sagot :

given:

[tex] \frac{1}{3} [/tex]

to find:

the probability of the coin.

solution:

lie:

[tex]( \frac{1}{3} ) \times ( \frac{1}{3}) \times ( \frac{1}{3} )[/tex]

[tex] = \frac{1}{27} [/tex]

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

truth:

[tex]( \frac{2}{3} ) \times ( \frac{2}{3} ) \times ( \frac{2}{3} )[/tex]

[tex] = \frac{8}{27} [/tex]

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

probability:

[tex]( \frac{8}{27} ) \div (( \frac{8}{27} ) + ( \frac{1}{27} ))[/tex]

[tex] = \frac{8}{9} [/tex]

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

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