# Caesar's Last Breath - a math question



## T_R_Oglodyte (May 14, 2010)

This being a math problem, you need to prepare yourself.  So before we go further take a deep breath help ensure you've got a clear mind.

Ready then??  OK - let's get started.

According to Bill the Bard, after Caesar was stabbed, he gasped and  then uttered, "Et tu, Brute?" with that last breath.

So, here's the question.  What is the probability that that deep breath you just took contains at least one of the same molecules that Caesar exhaled in that dying breath?

Assume that now, more than 2000 years later, those air molecules that Caesar exhaled are uniformly mixed throughout the earth's atmosphere, and that the vast majority of those molecules are still out there floating around and available to you to breathe.  (Since the most of the air that we breath is either inert nitrogen or oxygen that continually cycles in and out of the atmosphere, that's a reasonable assumption.)  Figure that you and Caesar each respire about 1/30th of a mole of air molecules with each breath.


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## scrapngen (May 14, 2010)

Oh, I hate moles...They make a mess of my garden. Oh, and my math brain gets weary when dealing with them...


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## Tom52 (May 14, 2010)

Water evaporates and returns to earth in the form of rain.  Then the water floats around and is sometimes consumed by animal life.  At least some of the water is expelled by the animal life in waste form.  The water evaporates from this waste form and eventually returns to earth in the form of rain.  The cycle repeats.

Why do I say this.........

Because I estimate the odds of just breathing some of Caesar's expelled air molecules from his last breath are about the same as just consuming in your morning cup of coffee, some of the water that passed thru Caesar's body the last time he visited the restroom before he was stabbed. 

So what's the answer?


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## tchr54 (May 14, 2010)

I think I like your logic.  Plus, since the water was basically earth bound, I would estimate the chances of ingesting Caesar's waste water to be a higher probability than his breath.  But, that's just a guess  
Have a great day!
Ed and Kay
Clinton Mo


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## Conan (May 14, 2010)

I'm answering this without Googling....

I remember from high school that a mole of a gas fills 22 liters and has 6 x 10^23 molecules so 1/30 of a mole is about 2 x 10^22 molecules per lungful.

So how many moles of air are there in the whole atmosphere? Air thins out pretty quickly so I'd say the volume of air compressed to a whole atmosphere is maybe 20 miles or say 30 km deep.

And I learned in high school that the volume of a sphere is 4/3 pi r^3

So taking the earth's radius as 4,000 miles or 6,000 km then the larger radius of the earth + atmosphere is (6,000 + 30). Which makes the volume of the atmosphere (compressed to 1 atm) 
4/3 * pi * (6,030^3 - 6,000^3) which is roughly 10^10 cubic km [I used my calculator for the cubing]
or 10^19 cubic meters or 10^22 liters or 5 x 10^20 moles or 3 x 10^44 molecules

Going back to Caesar, his last breath had about 2 x 10^22 molecules and the whole earth has about 2 x 10^44 molecules, so his contribution is one per 10^22. Since a lungful is also about 10^22 molecules we can say we get about 1 of Caesar's last-breath molecules each time we inhale.

Lots of rounding and assuming, but close enough?
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Edited to say - - I'm thinking there's less air in the atmosphere than I assumed - - once your plane is 5 miles high there's hardly any air outside, so my 30 km figure should have been more like 5 km, which means the volume of the earth's atmosphere is about 10^9 rather than 10^10
4/3 * pi * (6,005^3 - 6,000^3) = 10^9 cubic km. 
Which means Caesar's contribution is more than one per breath.


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## "Roger" (May 14, 2010)

I am working on the problem, but while I do let me offer a few random thoughts...

Medically, when you breathe in one of the molecules in question, you are said to have suffered a Caesure.  Contrary to what most people expect, you are much more likely to suffer a Caesure at lower altitudes than higher ones.  That is because at the higher altitudes the air is thinner and there are fewer moles per breathe.  The Incas at Machu Picchu tried to make up for this deficiency by serving their chicken with mole sauce...

Now, let me see,... there are 6.02 times 10 to the 23rd atoms per mole ...

All this reminds me of a story that I saw in the newspaper years ago.  A farmer set out to get into the Guinness Book of Records by growing the world's largest strawberry.  After carefully infusing a plant with every kind of fertilizer known to man, he had grown a plant that bore a twenty-two pound strawberry.  A neighbor suggested that he had better have the strawberry insured in case there was a late frost.  The farmer asked who would insure a strawberry and was told that Lloyds of London would insure anything.  Well, he contacted Lloyds and was told that they would send out an agent to look over the situation.  Two days later a man showed up at the door and asked to see the strawberry.  When shown to him, the man grabbed the strawberry and started to run off.  The farmer was able to apprehend him and asked what was up.  The man admitted that he was not an insurance agent from Lloyds, but was actually a Shakepearean actor.  "I came to seize your berry, not to appraise it."

On that low note ....  (6.02 times 10 to the 23rd divided by ....)


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## dmbrand (May 14, 2010)

So based on Conan's arithmetic, I will go with 100% probability. 

Also, we don't have many moles in our garden, but we have voles that munch on grass under the snow drifts some winters.  Makes a mess of the lawn.


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## nkldavy (May 14, 2010)

*My Brother Snoopy ...*

... tells me that Pig Pen  has some of Ceasar's dirt on him.

Uncle Davey


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## cgeidl (May 14, 2010)

*Caesar's Last breath for sale!!*

Everytime I exhale  the Exhalation with Ceasar's air is bottled. We are marketing and selling bottles containing the last breath of Caesar for only $29.95. Free shipping is included for TUB Suckers. Oops members only.


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## teepeeca (May 14, 2010)

*cgeidl*

What is a "TUB" Sucker???

Tony


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## Karen G (May 14, 2010)

Thanks for all the laughs. There are some mighty fine comedians on TUG.


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## hvacrsteve (May 14, 2010)

I finally figured it ouuuuuuuuuuuuut!

It is 1 in 14,456,236,345,567,045 not counting the air that is over the oceans surface since we don't occupy the oceans on a static basis!


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## Timeshare Von (May 15, 2010)

So when do we get the answer Steve?

My head wanted to explode even considering the "math" so my answer is easy . . . 100% . . . or 0%  

Von


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## T_R_Oglodyte (May 15, 2010)

Timeshare Von said:


> So when do we get the answer Steve?
> 
> My head wanted to explode even considering the "math" so my answer is easy . . . 100% . . . or 0%
> 
> Von



This seems like a good time; it's been out there long enough for people to take a shot at it.

The answer is that there's more than a 99% chance that each breath you take includes one or more molecules that was in Caesar's last breath.

****

First, let's say there are T molecules of air in the world.  Now lets say that A is the number of molecules of air in Caesar's last breath.

Then the probability that any given molecule of air you inhale is one that Caesar exhaled in that breath is A/T.  By extension, the probability that any single molecule you inhale was *not* part of Caesar's last breath is 1-A/T.

In you inhale two molecules, the probability that *both* molecules were not part of Caesar's last breath is:

(1-A/T)*(1-A/T)​
For three molecules the calculation becomes:

(1-A/T)*(1-A/T)*(1-A/T)​
Using the "^" character to indicate an exponent, as is done in Excel formulas, we can write the formula for n molecules as follows:

(1-A/T)^n​
That is the probability that *none* of the molecules were from Caesar's last breath.  If we let P be the probability that *at least one* of the molecules was from Caesar's last breath then:

P = 1 - (1-A/T)^n​
Now a human breath typically contains about 2.2x10^22 molecules (about one-third litre of air).  As Conan so aptly calculated, the world's atmosphere contains about 10^44 molecules of air.

So the probability that you respired at least one molecule from Caesar's dying breath is:

P = 1- {1 - [(2.2x10^22)/(1x10^44)^2.2x10^22]}​
Doing this calculation yields a result that is more that 99%.


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## Karen G (May 15, 2010)

There is no part of that explanation that I can understand. But, I'll take your word for it.  Math that goes beyond balancing my checkbook is way over my head. But, I'm glad their are minds that can grasp all this so that I don't have to.

The whole idea, though, of breathing air that has been breathed before is just another amazing fact about our unique planet and all that went into its creation.


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## T_R_Oglodyte (May 15, 2010)

So let's try a similar question.  The calculation approach is very similar to the Caesar's last breath question.

Let's say you are in Times Square on New Years Eve, waiting for the ball.  You begin asking random people in the crowd what their birthdate is (month and day).

How many people would you need to ask before there is a 50% chance that you would get a matching birthday?  (Use a 365 day calendar and ignore Feb 29). 

This problem is even more interesting if you first make a guess as to what you think the number would be before you work out an answer.


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## SpikeMauler (May 15, 2010)

I like eggs...


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## Rose Pink (May 15, 2010)

*Resident Statistician*

I'm surprised PJRose has not chimed in on this thread.  If anyone would know, it would be her.


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## pjrose (May 15, 2010)

Rose Pink said:


> I'm surprised PJRose has not chimed in on this thread.  If anyone would know, it would be her.



As soon as I saw the thread title, I thought Oh Goodie, a Math Problem!  When I read it I thought there wasn't enough information, such as what's a mole, how much air do we breathe, and how much air is in the atmosphere.  

I figured the answer - like that to the Birthday Problem - was a very high probability, otherwise it wouldn't be a very interesting question.

Troggy's very lucid explanation (Sorry Karen!) takes me off the hook!  

And I like eggs too.


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## easyrider (May 15, 2010)

Chance = 1

Probalbility = ?


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## "Roger" (May 15, 2010)

T_R_Oglodyte said:


> ....In you inhale two molecules, the probability that *both* molecules were not part of Caesar's last breath is:
> 
> (1-A/T)*(1-A/T)​
> ....


Tecnically not correct.  The chances of two molecules not being one of Caesar's is

(1-A/T)*[1-A/(T-1)]​etc.


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## Timeshare Von (May 15, 2010)

T_R_Oglodyte said:


> So let's try a similar question.  The calculation approach is very similar to the Caesar's last breath question.
> 
> Let's say you are in Times Square on New Years Eve, waiting for the ball.  You begin asking random people in the crowd what their birthdate is (month and day).
> 
> ...



It's like 24 people, I seem to recall from a prior discussion . . . or maybe it is with 24 people, the probably is like 80% to have two with the same BD???


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## Passepartout (May 15, 2010)

I've heard it's some number around 30 more or less. Seemed like there were always kids in my school classes who shared b-days. This is hardly the scientific and mathematically accurate description that describes _how_ this comes to be. Just anecdote.

Jim Ricks


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## scrapngen (May 15, 2010)

I seem to remember this from high school math - statistics unit - as well.
Although it seems to me that it was to see how high the number had to be to be close to 100%. And I believe the number was around 30 to have something like a 98% chance of two people sharing the same birthday. 

I could be wrong - as I can no longer do the math in my head, and not sure I could do it on paper.


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## T_R_Oglodyte (May 16, 2010)

scrapngen said:


> I seem to remember this from high school math - statistics unit - as well.
> Although it seems to me that it was to see how high the number had to be to be close to 100%. And I believe the number was around 30 to have something like a 98% chance of two people sharing the same birthday.
> 
> I could be wrong - as I can no longer do the math in my head, and not sure I could do it on paper.



It's 23 people to reach a 50% probability of having at least one pair of birthdays match up.  A 90% probability occurs at 41 people.  The 99th percentile occurs at 57 people.


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## scrapngen (May 17, 2010)

Curses! foiled again - my memory is only good when it doesn't require numerical accuracy... ok, that's probably no longer true either LOL!! and I haven't even hit 50. sigh...


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## pjrose (May 18, 2010)

scrapngen said:


> Curses! foiled again - my memory is only good when it doesn't require numerical accuracy... ok, that's probably no longer true either LOL!! and I haven't even hit 50. sigh...



If you have kids, it's THEIR age, not yours, that affects memory loss and other brain damage.  And the destruction of your brain cells has an exponential relationship to their age and how many crises they have experienced.  Believe me, I know....now what was the topic of this post again?


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## Rose Pink (May 18, 2010)

pjrose said:


> If you have kids, it's THEIR age, not yours, that affects memory loss and other brain damage. And the destruction of your brain cells has an exponential relationship to their age and how many crises they have experienced. Believe me, I know....now what was the topic of this post again?


_And_ . . . if you are caring for a loved one with dementia, it is contagious and your memory suffers even more.  (I am referring to other forms of dementia, not just teenage dementia, which is steeped in hormones.)


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