According to the video above, Bell's Inequality theorem shows that two fictional characters, Bob and Alice should get the same results more than 33% of the time. The actual probability of them getting the same outcome is 25%. But why 25%?
Since the particles Bob and Alice possess are entangled, and since Bob and Alice are not, it stands to reason that the probability of Bob and Alice being in sync would be less than expected.
Suppose Bob and Alice each randomly choose one of two polarizers: up or down. They compare notes hoping for a match. Next, they each send one of the entangled particles through their respective polarizers. Each check the spin of his/her particle to see if it is up or down. They compare notes again, hoping for another match.
If the particles weren't entangled the experiment would yield these possible outcomes:
The above outcomes show that Bob and Alice are in sync 50% of the time (which is greater than 33%). Four out of eight outcomes show their particles have the same spin or Alice and Bob picked the same polarizer.
Now here are the possible outcomes if the particles are entangled with opposite spin:
As you can see the probability falls from 50%. Alice's particle is no longer independent of Bob's. If Bob's particle's spin is up, Alice's is down, and vice versa. There are no longer up-up or down-down outcomes.
