Breathe in and then exhale. What is the probability that at least one of the molecules you inhaled was one of the molecules from Abraham Lincoln’s final breath? This is a well-defined event, and so it does have a probability. The question is how likely is this to occur? Pause for a moment and think what number sounds reasonable before reading any further.

### Assumptions

Let’s begin with identifying a few assumptions.

These assumptions will help in justifying certain steps in our calculation of this probability. We assume that since Lincoln’s death over 150 years ago the molecules from his last breath are spread out uniformly around the world. A second assumption is that most of these molecules are still part of the atmosphere, and able to be inhaled.

It’s worthwhile to note at this point that these two assumptions are what is important, not that the person we are asking the question about. Lincoln could be replaced with Napoleon, Gengis Khan or Joan of Arc. As long as enough time has passed to diffuse the final breath of a person, and for the final breath to escape into the surrounding atmosphere, the following analysis will be valid.

### Uniform

Start by selecting a single molecule. Suppose there are a total of *A* molecules of air in the world’s atmosphere. Furthermore, suppose that there were *B* molecules of air exhaled by Lincoln in his final breath.

By the uniform assumption, the probability that a single molecule of air that you inhale was part of Lincoln’s last breath is *B*/*A*. When we compare the volume of a single breath to the volume of the atmosphere, we see that this is a very small probability.

### Complement Rule

Next we use the complement rule.

The probability that any particular molecule that you inhale was not part of Lincoln’s last breath is 1 - *B*/*A*. This probability is very large.

### Multiplication Rule

Up until now we only consider one particular molecule. However, one’s final breath contains many molecules of air. Thus we consider several molecules by using the multiplication rule.

If we inhale two molecules, the probability that neither were part of Lincoln’s last breath is:

(1 - *B*/*A*)(1 - *B*/*A*) = (1 - *B*/*A*)^{2}

If we inhale three molecules, the probability that none were part of Lincoln’s last breath is:

(1 - *B*/*A*)(1 - *B*/*A*)(1 - *B*/*A*) = (1 - *B*/*A*)^{3}

In general, if we inhale *N* molecules, the probability that none were part of Lincoln’s last breath is:

(1 - *B*/*A*)^{N}.

### Complement Rule Again

We use the complement rule again. The probability that at least one molecule out of *N* was exhaled by Lincoln is:

1 - (1 - *B*/*A*)^{N}.

All that remains is to estimate values for *A, B* and *N*.

### Values

The volume of the average breath is about 1/30 of a liter, corresponding to 2.2 x 10^{22} molecules. This gives us a value for both *B* and *N*. There are approximately 10^{44} molecules in the atmosphere, giving us a value for *A*. When we plug these values into our formula, we end up with a probability that exceeds 99%.

Each and every breath that we take is almost certain to contain at least one molecule from Abraham Lincoln’s final breath.