What are the odds?
Consider that each person who has ever lived on this planet had two parents. If we assume for the sake of simplicity that a persons parents are on the average 20 years old when the person is born, we can draw a family tree with that person at the top, and each generation 20 years below it, thus:
We can see a progression here. With each generation, the number of ancestors doubles every 20 years. The first generation has 2, the second generation has 4, etc.
We can create a formula to tell us how many ancestors we have at each generation (assuming, for this exercise, that each of our ancestors is a different person). If Gen (G) is the number of the generation (0 being the current), and Num (N) is the number of ancestors, then our formula looks like this:
In table format,
(Note that the Yrs column also represents the age of that generation.)
Let's suppose we want to find out how many ancestors we had 100 years ago. Since each generation is 20 years, we divide 100 by 20 to get the number of generations, or 5.
The formula now looks like this:
So here is a table for a few interesting years
This shows that the number of ancestors we had 100 years ago, in the early 1900s, was 32.
It also shows that each of us had over 33 million ancestors if we go back just 500 years to the 16th century CE. How about 2000 years ago? Hold on to your hat!
This says that each of us had a gazillion ancestors (technically, 1.3 nonillion) if we go back just 2,000 years.
How is this possible? There weren't that many people around then. There aren't even that many people around now. I'll come back to this later.
Let me introduce a little simplifying terminology so we can write big numbers without all those digits from now on.
We may recall from our school days that 1,000 is 10 cubed or "10 to the power of 3." If we write "10 to the power of" as "e" (for exponent), then "e3" means "10 to the power of 3" or 1,000. We may also recall that 1,000 x 1,000 = 1,000,000. Writing it in our shorthand, then, e3 x e3 = 1,000,000. But 1,000,000 is also 10 to the power of 6. So e3 x e3 = e6. So we can see that adding the exponents is the same as multiplying the numbers. You can check this on various other powers of 10 if you like. Conveniently, you can also reconstruct the number by just writing a 1 followed by the number of zeroes matching the exponent. (E.g., e5 = 100,000)
You can see that some pretty big numbers can be easily represented in this manner.
Redoing our table from above and adding a few more, we get
Now lets add a few more points to the table.
You can see that the number of ancestors 6,000 years ago is 2 followed by 90 zeroes!
I find it interesting to look at these numbers in a slightly different sense. If you think about it, this says that in the last 6,000 years, 2.0e90 people had to pair up in one particular way in order for you to be conceived. In other words, the chances (odds) of you being born were 1 in 2.0e90 just 6,000 years ago. Yet, here you are.
I don't have access to the referenced work, but this statement appears on one of the Intelligent Design sites (http://intelligentdesign.org/odds/odds.htm).
I guess Dr. Borel never thought about his own heritage.
Another thing to ponder is just how large these numbers can get, as we keep doubling them every 20 years going into the past. Let's take it back further ...
So even if you like the "10 to the power of 200" odds as being impossible, you can see that just 20,000 years ago (1,000 generations ago) the odds were much longer than that. If you are of european stock, there is evidence of humans inhabiting Europe over 700,000 years ago.
Let's go back now to the quandary that there weren't that many people around back then, so what gives? The answer is that if we go back and look at a family tree of who these people are, we will find that they indeed are not all unique individuals. In fact, the further you go back in your tree, the more duplication you will find. Essentially, you will find that alot of cousins married (or at least mated with) each other, whether they knew it or not. There are other possibilities, of course, including siblings and transgenerational coupling, although most cultures forbid incest these days.
The fact is that there had to be that many people, unique or not -- after all it does take one of each sex to conceive a child -- it is just that some people could show up at more than one point in your family tree.
Yrs Gen Num
0 0 1 kid
|
+-----+-----+
| |
20 1 2 Mom Dad
| |
+---+---+ +---+---+
| | | |
40 2 4 Gma Gpa Gma Gpa
| | | |
We can see a progression here. With each generation, the number of ancestors doubles every 20 years. The first generation has 2, the second generation has 4, etc.
We can create a formula to tell us how many ancestors we have at each generation (assuming, for this exercise, that each of our ancestors is a different person). If Gen (G) is the number of the generation (0 being the current), and Num (N) is the number of ancestors, then our formula looks like this:
Num = 2 to the power of Gen
In table format,
Gen | Num | Yrs
-----+-------+-----
0 | 1 | 0 us
1 | 2 | 20 parents
2 | 4 | 40 grandparents
3 | 8 | 60 great grandparents
4 | 16 | 80 great great grandparents
(Note that the Yrs column also represents the age of that generation.)
Let's suppose we want to find out how many ancestors we had 100 years ago. Since each generation is 20 years, we divide 100 by 20 to get the number of generations, or 5.
The formula now looks like this:
Num = 2 to the power of Yrs divided by 20
So here is a table for a few interesting years
Yrs | Gen | Number
-----+-----+-------------
0 | 1 | 1 us
80 | 4 | 16 grandparents
100 | 5 | 32 great great great grandparents
120 | 6 | 64
200 | 10 | 1,024
300 | 15 | 32,768
500 | 25 | 33,554,432
This shows that the number of ancestors we had 100 years ago, in the early 1900s, was 32.
It also shows that each of us had over 33 million ancestors if we go back just 500 years to the 16th century CE. How about 2000 years ago? Hold on to your hat!
Yrs | Gen | Number
------+-----+------------------------------------------
500 | 25 | 33,554,432
1,000 | 50 | 1,125,899,906,842,624
1,500 | 75 | 37,778,931,862,957,161,709,568
2,000 | 100 | 1,267,650,600,228,229,401,496,703,205,376
This says that each of us had a gazillion ancestors (technically, 1.3 nonillion) if we go back just 2,000 years.
How is this possible? There weren't that many people around then. There aren't even that many people around now. I'll come back to this later.
Let me introduce a little simplifying terminology so we can write big numbers without all those digits from now on.
We may recall from our school days that 1,000 is 10 cubed or "10 to the power of 3." If we write "10 to the power of" as "e" (for exponent), then "e3" means "10 to the power of 3" or 1,000. We may also recall that 1,000 x 1,000 = 1,000,000. Writing it in our shorthand, then, e3 x e3 = 1,000,000. But 1,000,000 is also 10 to the power of 6. So e3 x e3 = e6. So we can see that adding the exponents is the same as multiplying the numbers. You can check this on various other powers of 10 if you like. Conveniently, you can also reconstruct the number by just writing a 1 followed by the number of zeroes matching the exponent. (E.g., e5 = 100,000)
You can see that some pretty big numbers can be easily represented in this manner.
Redoing our table from above and adding a few more, we get
Yrs | Gen | Number
------+-----+-----------
0 | 1 | 1
80 | 4 | 16
100 | 5 | 32
120 | 6 | 64
200 | 10 | 1.0e3
300 | 15 | 3.3e4
500 | 25 | 3.4e7
1,000 | 50 | 1.1e15
1,500 | 75 | 3.8e22
2,000 | 100 | 1.3e30
Now lets add a few more points to the table.
Yrs | Gen | Number
------+-----+-----------
1,000 | 50 | 1.1e15
1,500 | 75 | 3.8e22
2,000 | 100 | 1.3e30
3,000 | 150 | 1.4e45
4,000 | 200 | 1.6e60
5,000 | 250 | 1.8e75
6,000 | 300 | 2.0e90
You can see that the number of ancestors 6,000 years ago is 2 followed by 90 zeroes!
I find it interesting to look at these numbers in a slightly different sense. If you think about it, this says that in the last 6,000 years, 2.0e90 people had to pair up in one particular way in order for you to be conceived. In other words, the chances (odds) of you being born were 1 in 2.0e90 just 6,000 years ago. Yet, here you are.
I don't have access to the referenced work, but this statement appears on one of the Intelligent Design sites (http://intelligentdesign.org/odds/odds.htm).
It was Dr. Emile Borel who first formulated the basic Law of Probability which states that the occurrence of an event where the chances are beyond 1 chance in 10 to the 50th power (the 200th power is used for scientific calculations), is an event which we can state with certainty will never happen, regardless of the time allotted or how many opportunities could exist for the event to take place.(Emile Borel, Probabilities and Life, Dover 1962, chapters 1-3)
I guess Dr. Borel never thought about his own heritage.
Another thing to ponder is just how large these numbers can get, as we keep doubling them every 20 years going into the past. Let's take it back further ...
Yrs | Gen | Number
---------+--------+-----------
1,000 | 50 | 1.1e15
2,000 | 100 | 1.3e30
5,000 | 250 | 1.8e75
10,000 | 500 | 3.3e150
20,000 | 1,000 | 1.1e301
50,000 | 2,500 | 3.8e752
100,000 | 5,000 | 1.4e1505
200,000 | 10,000 | 2.0e3010
500,000 | 25,000 | 5.6e7525
1,000,000 | 50,000 | 3.2e15051
So even if you like the "10 to the power of 200" odds as being impossible, you can see that just 20,000 years ago (1,000 generations ago) the odds were much longer than that. If you are of european stock, there is evidence of humans inhabiting Europe over 700,000 years ago.
Let's go back now to the quandary that there weren't that many people around back then, so what gives? The answer is that if we go back and look at a family tree of who these people are, we will find that they indeed are not all unique individuals. In fact, the further you go back in your tree, the more duplication you will find. Essentially, you will find that alot of cousins married (or at least mated with) each other, whether they knew it or not. There are other possibilities, of course, including siblings and transgenerational coupling, although most cultures forbid incest these days.
The fact is that there had to be that many people, unique or not -- after all it does take one of each sex to conceive a child -- it is just that some people could show up at more than one point in your family tree.
posted by John Kingman at 10:41 PM
0 Comments:
Post a Comment
<< Home