Schrödinger’s Person A – Explanation

Problem Statement

"Until the box is opened and the contents are observed, the cat exists in a state where 'being alive' and 'being dead' are superimposed—each accounting for half of its reality."

This is the world's most famous thought experiment, devised by the physicist Schrödinger to illustrate a peculiar rule of quantum mechanics: the state of an object remains undetermined—existing as a superposition, much like a wave—until it is observed.

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Person A, who is in poor health and has an extreme tendency to lose interest in things, is currently shut away in a room with the door firmly closed.

From the outside, it is impossible to tell—even for their own parents—whether Person A is currently "awake" (ON) or "asleep" (OFF). (Perhaps that’s not strictly true in reality, but according to quantum mechanics, it is unknowable.)

(To put it simply: if Person A were in a room that was completely soundproof, you would have no way of knowing whether they were awake or asleep.)

Applying the rules of quantum mechanics to Person A, it follows that until you open the door and "observe" the room, Person A exists in a state where "being awake" and "being asleep" are superimposed within the room.

Person A’s current, uncanny state ($\psi$ψ) can be expressed by the following "state equation":

$\psi=\alpha\vert\mathrm{ON}\rangle+\beta\vert\mathrm{OFF}\rangle$ψ=α∣ON⟩+β∣OFF⟩

Here, $|\mathrm{ON}\rangle$∣ON⟩ represents the state of being awake, and $|\mathrm{OFF}\rangle$∣OFF⟩ represents the state of being asleep—these symbols denote the actual states that are currently superimposed. (Think of them as placeholders for potential outcomes; it might be easiest to view them as representing "future states.")

$\alpha$α and $\beta$β are components related to probability—essentially representing the "strength" or "weight" of each state at that given moment.

However, under the rules of quantum mechanics, these "strength" values ​​cannot be taken as probabilities directly. There is a curious rule stating that the actual probability is obtained by squaring these values. (Please look up the details yourself; I don’t know them either.)

$\alpha^2+\beta^2=1$α2+β2=1

My friend B analyzed a faint sound heard from outside a door and determined that $\alpha_1$α1​—the value representing the basis for the probability that A is awake—is $0.8432$0.8432.

Assume that this value remains constant until the door is opened and the situation inside is checked.

When the door is opened and A’s state is observed, calculate $\beta_1$β1​—the value representing the basis for the probability that A is asleep—derived from the probability that A is asleep.

Note: Assume that no observation took place at the time the sound was heard.

Note: While $\alpha$α and $\beta$β are typically calculated as complex numbers, for the sake of simplification, please perform the calculations using decimal numbers.

Constraints

$\alpha_1=0.8432$α1​=0.8432

Input Format

Provide the value of $\beta_1 \times 10000$β1​×10000 as your answer.