## Have You Seen A Half-Balloon?

Balloons come in all shapes and sizes but the shape the comes to mind when I hear the word balloon is that of a spherical or round shape. Given a spherical balloon, create an imaginary partition that divides the balloon into left and right hemispheres. This effectively places the air molecules into left and right. The number of molecules in the left is more or less the same as the number of molecules on the right. This is what we see in real life.

As we pump air into the balloon, a given air molecule can randomly go left or right of the partition. If all air molecules for some reason go left (or right), then we have a situation called a “half-balloon”.

So why aren’t we seeing half-balloons? It’s because that situation is highly improbable. For illustration purposes, imagine there are 6 air molecules. Each molecule has two equally likely choices, be on the left or on the right. Now imagine each air molecule has chosen it’s “side”. A possible combination or configuration (a term we will use moving forward) will be 3 molecules choose Left and 3 choose Right:

L  L  L  R  R  R


Where L stands for Left and R stands for Right.

In fact, the number of such configurations is equal to

$\displaystyle \underbrace{2\times 2\times \ldots \times 2}_{\text{6 times}} = 2^6 = 64$

We enumerate below the list of possible air molecule configurations:


# This is the code in R
# x=c("L","R")
# expand.grid(m1=x,m2=x,m3=x,m4=x,m5=x,m6=x)
# d1=expand.grid(m1=x,m2=x,m3=x,m4=x,m5=x,m6=x)
m1 m2 m3 m4 m5 m6
1   L  L  L  L  L  L
2   R  L  L  L  L  L
3   L  R  L  L  L  L
4   R  R  L  L  L  L
5   L  L  R  L  L  L
6   R  L  R  L  L  L
7   L  R  R  L  L  L
8   R  R  R  L  L  L
9   L  L  L  R  L  L
10  R  L  L  R  L  L
11  L  R  L  R  L  L
12  R  R  L  R  L  L
13  L  L  R  R  L  L
14  R  L  R  R  L  L
15  L  R  R  R  L  L
16  R  R  R  R  L  L
17  L  L  L  L  R  L
18  R  L  L  L  R  L
19  L  R  L  L  R  L
20  R  R  L  L  R  L
21  L  L  R  L  R  L
22  R  L  R  L  R  L
23  L  R  R  L  R  L
24  R  R  R  L  R  L
25  L  L  L  R  R  L
26  R  L  L  R  R  L
27  L  R  L  R  R  L
28  R  R  L  R  R  L
29  L  L  R  R  R  L
30  R  L  R  R  R  L
31  L  R  R  R  R  L
32  R  R  R  R  R  L
33  L  L  L  L  L  R
34  R  L  L  L  L  R
35  L  R  L  L  L  R
36  R  R  L  L  L  R
37  L  L  R  L  L  R
38  R  L  R  L  L  R
39  L  R  R  L  L  R
40  R  R  R  L  L  R
41  L  L  L  R  L  R
42  R  L  L  R  L  R
43  L  R  L  R  L  R
44  R  R  L  R  L  R
45  L  L  R  R  L  R
46  R  L  R  R  L  R
47  L  R  R  R  L  R
48  R  R  R  R  L  R
49  L  L  L  L  R  R
50  R  L  L  L  R  R
51  L  R  L  L  R  R
52  R  R  L  L  R  R
53  L  L  R  L  R  R
54  R  L  R  L  R  R
55  L  R  R  L  R  R
56  R  R  R  L  R  R
57  L  L  L  R  R  R
58  R  L  L  R  R  R
59  L  R  L  R  R  R
60  R  R  L  R  R  R
61  L  L  R  R  R  R
62  R  L  R  R  R  R
63  L  R  R  R  R  R
64  R  R  R  R  R  R


Looking at the data above, we can see that there are 6 configurations that contain only 1 “L”:

# This is the code in R
# cc=c()
# for(i in 1:64){
#  tmp=d1[i,]
#  cc=c(cc,sum(tmp == "L"))
# }
# d1$count = cc # d1[d1$count == 1, ]
m1 m2 m3 m4 m5 m6 count
32  R  R  R  R  R  L     1
48  R  R  R  R  L  R     1
56  R  R  R  L  R  R     1
60  R  R  L  R  R  R     1
62  R  L  R  R  R  R     1
63  L  R  R  R  R  R     1


This means that the probability of getting a configuration having one molecule on the left and 5 molecules on the right is:

$\displaystyle \text{Probability of 1 molecule Left and 5 molecules Right} = \frac{6}{2^6} = 0.093750$

If we continue summarizing our data so that we get the number of combinations containing 2, 3, 4, 5, and 6 “L”s, we can generate what is known as a probability distribution of air molecules on the left hemisphere:

# cc=c()
# for(i in 0:6){
#  cc=c(cc,length(attributes(d1[d1$count == i, ])$row.names))
# }
# data.frame(X=0:6,count=cc,probability=cc/2^6)
X count probability
1 0     1    0.015625
2 1     6    0.093750
3 2    15    0.234375
4 3    20    0.312500
5 4    15    0.234375
6 5     6    0.093750
7 6     1    0.015625


The graph of this probability distribution is shown below:

# Here is the code that generated the plot above
barplot(cc/2^6,names.arg=0:6,main="Probability Distribution \nNumber of Air Molecules on the Left Hemisphere")


The graph above tells us that the most probable configuration of balloon molecules is that half of them are on the right and the other half on the left as shown by the high probability of the value X = 3. It also tells us the probability of all molecules choosing the Left side equal to 0.015625. When the number of air molecules is very large, this probability will turn out to be extremely small, as we will show below.

## The Mathematics

At this point, manually counting the number of rows will not be practical anymore for a large number of molecules. We need to use some mathematical formula to compute the combinations. We don’t really care which molecules chose left or right. We just care about the number of molecules on the left. Given N molecules, there are $2^N$ possible configurations of air molecules. Of these configurations, there are

$\displaystyle {N \choose m } = \frac{N!}{m! (N-m)!}$

combinations that have $m$ molecules on the left. Therefore, the probability of a configuration having $m$ molecules on the left is

$P(m) = \displaystyle \frac{\displaystyle {N\choose m}}{\displaystyle 2^N}$

This is a probability density function since

$\displaystyle \sum_{m=0}^N P(m) = \displaystyle \sum_{m=0}^N \frac{\displaystyle {N\choose m}}{\displaystyle 2^N} = 1$

To show this, we will use the Binomial Theorem

$\displaystyle \sum_{m=0}^N {N \choose m} x^m = (1+x)^N$

If we let $x=1$, the Binomial Theorem gives us

$\displaystyle \sum_{m=0}^N {N \choose m} = (1+1)^N = 2^N$

Therefore

$\begin{array}{rl} \displaystyle \sum_{m=0}^N P(m) &= \displaystyle \sum_{m=0}^N \frac{\displaystyle {N\choose m}}{\displaystyle 2^N}\\ &= \displaystyle \frac{1}{2^N} \sum_{m=0}^N {N\choose m}\\ &= \displaystyle \frac{1}{2^N} 2^N\\ &= 1 \end{array}$

## A Mole of Air

One mole of air contains $6.022 \times 10^{23}$. This means the probability of that all molecules are on the left side of the balloon is

$\displaystyle \frac{1}{\displaystyle 2^{6.022 \times 10^{23}}} < 1/1024^{10^{22}} < 1/(10^3)^{10^{22}}=10^{-30,000,000,000,000,000,000,000}$

This is a very small number such that it contains 30 million trillion zeros to the right of the decimal point. When you write this zeros on a piece of paper with a thickness of 0.05 millimeters, you would need to stack it up to a height 74 times the distance of earth to pluto in kilometers!