The city of Roboville has been invaded by robots, monkeys, and dancers, who will dance if the beat's funky. Some of the invaders can be two of them (e.g. both monkeys and robots) and some can be all three. But they all look like normal humans, and you have absolutely no clue how the invaders are and their types.
But you know that among all the residents (invaders and non-invaders), 1 in 2 are robots, 1 in 3 are monkeys, and 1 in 4 are dancers (all independent).
It is known that everytime they say three sentences, robots lie on their first sentence, monkeys on their second, dancers on their third, and non-invaders always tell the truth. So robots who are also monkeys will lie twice then tell the truth.
The Roboville government has set up a reCaptcha test to get rid of all the invaders. Only non-invaders (those who are not robots, monkeys or dancers) will pass the test.
Suppose someone in Roboville says, 'I'm not a monkey. I will not dance even if the beat's funky. I'm not a robot', what is the chance of them passing the reCaptcha?
Suppose the current population of Roboville is 12,000,000 (including invaders), which is divided into six equal groups,
in which everyone from each group says a different combination of the sentences 'I'm not a robot, I'm not a monkey, I will not dance even if the beat's funky'
(e.g. One group says 'not a robot, not a monkey, not dance', another says 'not a robot, not dance, not a monkey' and so on),
what is the expected number of people who will pass the reCaptcha? (if not an integer, round off to three significant figures)
This puzzle is heavily based on Linkin Park's song When They Come For Me.