Thermal Physics - Probability

A sample of N atoms of helium gas is confined in a 1.0 cubic meter volume. The probability that none of the helium atoms is in a 10⁻⁶ cubic meter volume of the container is

A. 0
B. (10⁻⁶)^N
C. (1 − 10⁻⁶)^N
D. 1 − (10⁻⁶)^N
E. 1

(GR8677 #15)

Solution:

Total Probability: P = P₁ + P₂ = 1
P₁ = The probability that one atom is in a 10⁻⁶ cubic meter volume of the container
P₂ =  The probability that none of the helium atoms is in a 10⁻⁶ cubic meter volume of the container
→ P₂ = 1 − P₁

P₁ = 1/n
n = number of 10⁻⁶ m³ cubes in the 1 m³ volume
10⁻⁶ × n = 1 → n = 1/10⁻⁶ = 10⁶ cubes
→ P₁ = 1/n = 1/10⁶ = 10⁻⁶
→ P₂ = 1 − P₁ = 1 − 10⁻⁶
For N atoms: P₂ = (1 − 10⁻⁶)^N

Answer: C