Question

A) A closed box has 5 molecules and is then partitioned instantaneously into 4 equally sized compartments. What is the probability of there being 4 particles in the top left compartment?

B) Now imagine the same experiment as part A, but with 3 million molecules. What is the range of molecules that are in each partition 95% of the time? Hint: ±2 standard deviations is 95% of the normal distribution.

Answer 1

a)

here this is binomial distribution with parameter n=5 and p=1/4

therefore probability of there being 4 particles in the top left compartment =P(X=4)=⁵C₄(1/4)⁴(3/4)¹

=0.0146

b)

here expected number of particles in a partition =np=3*10⁶*0.25 =750000

and std deviation =sqrt(np(1-p))=sqrt(3*10⁶*0.25*0.75)=750

hence 95% range of molecules in each partition =estimated mean -/+ 2*std deviation

=748500 to 751500