The new shepherd boy was being instructed by the villagers on his job of keeping the sheep safe in their pasture outside the village.
The village statistician told the boy, "Do not cry out unless you see a wolf. That way, when we hear a cry, we will know a wolf has come, and we will run here to drive it away. That is, writing \( \rm P(A|B) \) for the probability of A given B, we have:" $$ \rm P(wolf|cry) = \frac{\rm P(cry|wolf) \times P(wolf)}{\rm P(cry)} = \frac{\rm 100\% \times P(wolf)}{\rm P(wolf)} = 100\% $$ Although the boy did not pay much attention to the statistician, he did watch the sheep carefully the first day, all by himself in the lonely pasture at the edge of the dark forest.
But the next day, bored and wanting excitement, the shepherd boy cried "Wolf!" even though no wolf had shown up. The villagers came running, and the boy laughed to see their consternation.
The village statistician frowned and told the boy, "Now we can no longer be sure that a wolf has come when we hear you cry. Assuming each day there is a 1 in 3 chance that a wolf will show up, and now a 1 in 2 chance that you will cry out despite no wolf showing up, I estimate that the next time you cry wolf the odds will be only 50:50 that a wolf really has shown up. That is:" $$\rm P(wolf | cry) = \frac{\rm P(cry|wolf) \times P(wolf) }{\rm P(cry)} = \frac{\rm P(cry|wolf) \times P(wolf) }{\rm P(cry|wolf) \times P(wolf) + P(cry|no~wolf) \times P(no~wolf)} $$ "Or, dividing the numerator into the numerator and the denominator:" $$\rm P(wolf | cry) = \left[ 1 + \frac{\rm P(cry | no~wolf ) \times P(no~wolf)}{\rm P(cry | wolf) \times P(wolf)} \right]^{-1} $$ "And plugging in our assumptions:" $$\rm P(wolf | cry) = \left[ 1 + \frac{\rm (1/2) \times (2/3)}{\rm 1 \times (1/3) } \right]^{-1} = \left[ 1 + \frac{1/3}{1/3} \right]^{-1} = \left[ 2 \right]^{-1} = \frac{1}{2} $$
The foolish shepherd boy could not follow the derivation, and repeated the trick again the next day, laughing even harder. The village statistician shook his head sadly and told the boy, "Now I estimate that the odds will be only 3 in 7 that a wolf really has shown up the next time you cry wolf."
The next day the wolf --- who had been hiding near the pasture and had paid careful attention to the statistician --- jumped into the pasture when the shepherd boy arrived with the sheep. The boy cried out and the villagers heard, but the statistician only sighed. "In my estimation, the probability that this is a trick is larger than the probability that a wolf has shown up." So he and the villagers did not run to the pasture, and the wolf devoured the sheep.
Moral: no matter what the prior probability,
a low enough likelihood makes for a low posterior probability.
Discussion points:
* What assumptions did the village statistician adopt after the second trick to derive the 3 in 7 probability mentioned above?
* Did the village statistician adopt a statistically significant probability threshold for ignoring the shepherd boy?
* Is the village statistician also just a liar, or a damn liar?
* The wolf knew math?! What are the odds?!?