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Probability of Picking in the Correct Order

Date: 4/12/96 at 20:21:7
Subject: I'm stumped.

Hello Dr.Math.

I need your help.  Someone once told me the answer to this 
probability question is 1 out of 10 to the 287th power - a very 
tiny number, so small that we can consider it as not "probable" 
even if we keep on trying every second for 4 billion years.  He 
also said that most mathematicians would dismiss an event if it 
has a probability of 1 out of 10 to the 50th power.  (Please also 
comment on this.)

Here is the probability question:

Out of a population of 200 different amino acids, what is the 
probability of selecting the correct 20 acids in the correct order 
on your first try? In other words, say the 200 acids are numbered 
from 1 to 200 and you want only the acids numbered 1 to 20.  The 
question is what is the probability that you will pick the correct 
20 acids in this order: 1, 2, 3, 4, ....20 ?  If you keep trying 
every second for the next 4 billions year, will you be able to get 
it right?

The second part of this question is, what then is the probability 
that you will pick the correct 20 in the specified order, 232 
times in a row?

I really appreciate your help.


Date: 4/27/96 at 18:47:9
From: Doctor Steven
Subject: Re: I'm stumped.

Well, here goes.

We have 20 amino acids which need to go in correct order, and we 
can only accomplish this in one way (pick the amino acids in 
order).  Our number of choices for the first amino acid is 200, 
the second 199, the third 198, . . . the twentieth 181.  So there 
are 200*199*198*. . .*181 different possibilities for what we can 
choose.  Only one is correct.  So we have a 1/(200*199*198* . . 
.*181) percent chance of getting it correct.  This is roughly 1/(4 
* 10^45), so if we choose one set every 20 seconds, we can get 3 
sets per minute, so we need 1.333 * 10^45, or 2.222 * 10^43 hours, 
or 9.26 * 10^41 days, or 2.535 * 10^39 years. Since billions are 
only 10^9 the chances are no.

Now the next question is what are the chances of doing this 232 
times in a row.  The chances are 
[1/(4 * 10^45)]^232 = [1/(4^232 * 10^10440)] = 
[1/(1.6 * 10^60)*(1.6 * 10^60)*(1.8 * 10^19)*(10^10440)] = 
[1/(4.6 * 10^10579)]. 
So the chances are pretty slim.  

And yes, when you get to things that have chances that are smaller 
than (1/10^50) I would consider a good idea to just say it's not 
going to happen.

Hope this helps.

-Doctor Steven,  The Math Forum

Date: 5/1/96 at 16:26:4
From: Anonymous
Subject: Re: I'm stumped.

Dear Dr. Math,

By this mathematical proof, why are there still people who do not 
believe in creation but still diametrically believe life came 
about in accordance to the evolution theory (aka big bang) ?

From my bio class, I learned that the building blocks of life are 
amino acids.  But not just any amino acids will do or any random 
combination of them will do.  Only the sets of 20 amino acids that 
have the probability of 1/ (4 * 10^45) of ever coming together 
correctly to form a simple protein molecule that can possibly 
originate life.

The big bang theory says the origin of earth/life began about 4 
billions years ago. Given even that length of time, the random 
choosing of the correct 20 amino acids to form a simple protein 
molecule couldn't mathematically prove to be possible.

And we have 232 different types of vital proteins in our 
human body.

Thanks again,


Date: 5/4/96 at 11:35:6
From: Doctor Steven
Subject: Re: I'm stumped.

This is by no means a rebuttal of your logic since I am no expert 
on either side of this argument. But here is where your logic 
might have some holes:

The chance of my winning the lottery is about 1 in 5 million.  
The chance of *someone* winning the lottery is about 95%.

The same sort of principle holds here:  the chance of earth being 
the place where life started is 1/4*10^45: the chance of life 
starting *somewhere* will be much greater.

Besides, the 232 proteins we have didn't just spontaneously occur 
- they were adapted from previous proteins to do a better job.  I 
mean how long does it take for a new protein to develop? If this 
new protein does a good job the organism lives longer and makes 
more different proteins.  

If you can refute these arguments (and probably many others) you 
will have a good case for creationism.

-Doctor Steven,  The Math Forum

Associated Topics:
High School Probability

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