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

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