Counter-intuitive thinking is one of the higher-order thinking skills that is difficult to teach, and usually arises when confronted with a real problem

During WWII, statistician Abraham Wald was asked to help the British decide where to add armor to their bombers. After analyzing the records, he recommended adding more armor to the places where there was no damage!

The RAF was initially confused.

Wald had data only on the planes that returned to Britain so the bullet holes that Wald saw were all in places where a plane could be hit and still survive.

The planes that were shot down were probably hit in different places than those that returned so Wald recommended adding armor to the places where the surviving planes were lucky enough not to have been hit.

British flew at night Americans during the day btw

Here’s another riddle: A man goes to a courthouse with an urgent message for a lawyer, who is in a meeting on floor 19, which cannot be interrupted. Five of seven elevators are out of service, and there is a 45 minute long line to get on an elevator.

The man resolves to walk up the stairs due to the urgency of this message. When he gets into the stairwell, he realizes that starting on floor 4 all the doors to the floors are locked due to security reasons.

Suddenly, he realizes what he must do, and five minutes later, using the elevator, he was on floor 19. How did he do it?

He got out of the stairwell on the 3rd floor, caught an elevator down, stayed in it, and went back up to the 19th floor, delivering the urgent message in the nick of time.

Parrondo's Paradox is a double shocker. Counter to common intuition, it is possible to mix two losing games into a winning combination.

Parrondo's Paradox was dreamed up in the 1990s by physicist Juan Manuel Rodriguez Parrondo. It spawned a whole new approach to games — specifically, a distrustful approach to games by those who were sure the odds were stacked in their favor.

The paradox is simple: two games, if played separately, will always result in you losing your shirt. They're played with a biased coin to make sure of it. If you switch off between them, though, you'll win a fortune.

Suddenly, your loss turns into a win.

The first game is simple and always the same. You flip a coin, knowing that the two-faced, lying, no-good cheater you're playing against has weighted it so that your chance of winning is not fifty-fifty. Instead your chance of winning is (0.5 - x), with x being whatever the cheater dared weight it with.

If you win, you get a dollar. If you lose, you lose a dollar. Since whenever "x" is more than zero you'll lose slightly more than you'll win, you are guaranteed to lose over the long run.

Your second game is played for the same stakes (win or lose a dollar) but two different ways. First, you look at the money you have in dollars and see if it's a multiple of three. If it isn't, out comes another biased coin, and it gives you odds of winning at (0.75 - x).

That means your chance of winning is seventy-five percent, minus whatever "x" was in the initial game. Well, that doesn't look too bad. Why is this a losing game?

If the money you have is not a multiple of three, then you play against the really bad coin. This coin is weighted so that your chance of winning is (0.10 - x). That is a less than a one-tenth chance of winning.

Since there are two times as many non-multiples of three than there are multiples of three, you will be playing with the coin that gives you an over-ninety percent chance of losing twice as much as the one that gives you a less-than seventy-five percent chance of winning. In other words, you will lose this game, and you will lose badly.

Tests have shown that, played one hundred times, either games results in lost money as long as "x" is bigger than zero. You can't beat those odds. But you can. Oh yes, you can. Switch between the games, playing the first one twice and then the next one twice, and you will win money.

It has been shown that, with x = 0.005, and with other values, depending on the sequence of the game, the winnings stack up. You don't even have stick to an orderly way of switching back and forth between the games.

 Randomly flip an (unbiased) coin, and you will still win in the long run. (There are many iterations of Parrondo's game.

Some of them have different values of "x," and in these the second game is played with different multiples, so you'll play with the first coin only if your winnings are a multiple of four or five or six and so on. There are many ways to win.)

https://www.cut-the-knot.org/ctk/Parrondo.shtml
http://datagenetics.com/blog/august22012/index.html
https://schooltutoring.com/help/food-for-thought-counter-intuitive-thinking/
https://www.reddit.com/r/AskReddit/comments/284lo9/what_is_a_good_example_of_something_that_is/
https://io9.gizmodo.com/5861287/parrondos-paradox-winning-two-games-youre-guaranteed-to-lose