How changing the basis of measurement produces a state which is not in the possible system states?

This is a Prologue taken from David McIntyre's Quantum Mechanics textbook (the link should lead you to the pdf version of the textbook) that I thought is pretty neat and shows the counter intuitive of QM. Also, section 1.1 describes the measurement phenomena of QM really nicely. I encourage you to read through it.

It was a dark and stormy night. Erwin huddled under his covers as he had done numerous times that summer. As the wind and rain lashed at the window, he feared having to retreat to the storm cellar once again. The residents of Erwin's apartment building sought shelter whenever there were threats of tornadoes in the area. While it was safe down there, Erwin feared the ridicule he would face once again from the other school boys. In the rush to the cellar, Erwin seemed to always end up with a random pair of socks, and the other boys teased him about it mercilessly.

Not that Erwin hadn't tried hard to solve this problem. He had a very simple collection of socks - black or white, for either school or play; short or long, for either trousers or lederhosen. After the first few teasing episodes from the other boys, Erwin had sorted his socks into two separate drawers. He placed all the black socks in one drawer and all the white socks in another drawer. Erwin figured he could determine an individual sock's length in the dark of night simply by feeling it, but he had to have them presorted into white and black because the apartment generally lost power before the call to the shelter.

Unfortunately, Erwin found that this presorting of the socks by color was ineffective. Whenever he reached into the white sock drawer and chose two long socks, or two short socks, there was a 50% probability of any one sock being black or white. The results from the black short drawer were the same. The socks seemed to have "forgotten" the color that Erwin had determined previously.

Erwin also tried sorting the socks into two drawers based upon their length, without regard to color. When he chose black or white socks from these long or short drawer, the socks had also "forgotten" whether they were long or short.

After these fruitless attempts to solve this probem through experiments, Erwin decided to save himself the fashion embarrassment, and he replaced his sock collection with a set of medium length brown socks. However, he continued to ponder the mysteries of the socks throughout his childhood.

After many years of daydreaming about the mystery socks, Erwin Schrodinger proposed his theory of "Quantum Socks" and become famous. And this is the beginning of the story of the quantum socks.

Farfetched?? You bet. But Erwin's adventure with his socks is the way quantum mechanics works.