Consiglio Devastations

Probability Level 5

Suppose every person on Earth was given a button which, when pressed, may either kill a random person, or just do nothing. If everyone pressed their respective buttons simultaneously,what is the expected number of people that would live?

Details and Assumptions:

Every button has an equal chance of being able to kill or not.
For this problem, there are exactly seven billion people on Earth.
It is possible that a person may be able to kill himself once he/she pressed the button.
Overlaps on who the buttons kill are allowed.
A $0.8$ person is not a person.

The answer is 4245714617.

1 solution

Efren Medallo
Mar 13, 2017

Hello Brilliant! It has been so long.

Let us scale down first the problem with $10$ people.

What are the odds that you'll die for each button push? That is $0.10 \times 0.5 = 0.05$ . Consequently, your odds of surviving that same button push would be $0.95$ .

For two independent button pushes, your chance of survival would be $(0.95)^2$ . For three, it would be $(0.95)^3$ , and so on. So for ten buttons, it would be $(0.95)^{10}$ .

Actually, the general model for determining the survival rate for $n$ buttons with individual success rate of $a$ (which, by the way, is $0.5$ in this problem) is

$(1 - \frac{a}{n})^n$

which, at sufficiently large values of $n$ , approaches

$e^{-a}$

So, there you have it. We'll have a survival rate of $\frac {100}{\sqrt{e}}$ %. Multiply that to the assumed total population, and that will give us the desired number which is $4, 245, 714, 617$ .

Consiglio Devastations

The answer is 4245714617.

1 solution

0 pending reports